大连理工大学优化方法上机大作业
2016年大连理工大学优化方法上机大作业
2016年理工大学优化方法上机大作业学院：专业：班级：学号：：上机大作业1：1.最速下降法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = steepest(x0,eps)gk = grad(x0);res = norm(gk);k = 0;while res > eps && k<=1000dk = -gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;gk = grad(xk);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=steepest(x0,eps)2.牛顿法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad2(x)g = zeros(2,2);g(1,1)=2+400*(3*x(1)^2-x(2));g(1,2)=-400*x(1);g(2,1)=-400*x(1);g(2,2)=200;endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = newton(x0,eps)gk = grad(x0);bk = [grad2(x0)]^(-1);res = norm(gk);k = 0;while res > eps && k<=1000dk=-bk*gk;xk=x0+dk;k = k+1;x0 = xk;gk = grad(xk);bk = [grad2(xk)]^(-1);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x1=newton(x0,eps)--The 1-th iter, the residual is 447.213595--The 2-th iter, the residual is 0.000000x1 =1.00001.00003.BFGS法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = bfgs(x0,eps)g0 = grad(x0);gk=g0;res = norm(gk);Hk=eye(2);k = 0;while res > eps && k<=1000dk = -Hk*gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;fa0=xk-x0;x0 = xk;go=gk;gk = grad(xk);y0=gk-g0;Hk=((eye(2)-fa0*(y0)')/((fa0)'*(y0)))*((eye(2)-(y0)*(fa0)')/((fa0)'*( y0)))+(fa0*(fa0)')/((fa0)'*(y0));res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;End>> clear>> x0=[0,0]';>> eps=1e-4;>> x=bfgs(x0,eps)4.共轭梯度法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star =CG(x0,eps)gk = grad(x0);res = norm(gk);k = 0;dk = -gk;while res > eps && k<=1000ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;g0=gk;gk = grad(xk);res = norm(gk);p=(gk/g0)^2;dk1=dk;dk=-gk+p*dk1;fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk; end>> clear>> x0=[0,0]'; >> eps=1e-4; >> x=CG(x0,eps)上机大作业2：function f= obj(x)f=4*x(1)-x(2)^2-12;endfunction [h,g] =constrains(x) h=x(1)^2+x(2)^2-25;g=zeros(3,1);g(1)=-10*x(1)+x(1)^2-10*x(2)+x(2)^2+34;g(2)=-x(1);g(3)=-x(2);endfunction f=alobj(x) %拉格朗日增广函数%N_equ等式约束个数?%N_inequ不等式约束个数N_equ=1;N_inequ=3;global r_al pena;%全局变量h_equ=0;h_inequ=0;[h,g]=constrains(x);%等式约束部分?for i=1:N_equh_equ=h_equ+h(i)*r_al(i)+(pena/2)*h(i).^2;end%不等式约束部分for i=1:N_inequh_inequ=h_inequ+(0.5/pena)*(max(0,(r_al(i)+pena*g(i))).^2-r_al(i).^2) ;end%拉格朗日增广函数值f=obj(x)+h_equ+h_inequ;function f=compare(x)global r_al pena N_equ N_inequ;N_equ=1;N_inequ=3;h_inequ=zeros(3,1);[h,g]=constrains(x);%等式部分for i=1:1h_equ=abs(h(i));end%不等式部分for i=1:3h_inequ=abs(max(g(i),-r_al(i+1)/pena));endh1 = max(h_inequ);f= max(abs(h_equ),h1); %sqrt(h_equ+h_inequ);function [ x,fmin,k] =almain(x_al)%本程序为拉格朗日乘子算法示例算法%函数输入：% x_al:初始迭代点% r_al：初始拉格朗日乘子N-equ:等式约束个数N_inequ:不等式约束个数?%函数输出% X：最优函数点FVAL:最优函数值%============================程序开始================================ global r_al pena ; %参数（全局变量）pena=10; %惩罚系数r_al=[1,1,1,1];c_scale=2; %乘法系数乘数cta=0.5; %下降标准系数e_al=1e-4; %误差控制围max_itera=25;out_itera=1; %迭代次数%===========================算法迭代开始============================= while out_itera<max_iterax_al0=x_al;r_al0=r_al;%判断函数?compareFlag=compare(x_al0);%无约束的拟牛顿法BFGS[X,fmin]=fminunc(alobj,x_al0);x_al=X; %得到新迭代点%判断停止条件?if compare(x_al)<e_aldisp('we get the opt point');breakend%c判断函数下降度?if compare(x_al)<cta*compareFlagpena=1*pena; %可以根据需要修改惩罚系数变量elsepena=min(1000,c_scale*pena); %%乘法系数最大1000disp('pena=2*pena');end%%?更新拉格朗日乘子[h,g]=constrains(x_al);for i=1:1%%等式约束部分r_al(i)= r_al0(i)+pena*h(i);endfor i=1:3%%不等式约束部分r_al(i+1)=max(0,(r_al0(i+1)+pena*g(i)));endout_itera=out_itera+1;end%+++++++++++++++++++++++++++迭代结束+++++++++++++++++++++++++++++++++ disp('the iteration number');k=out_itera;disp('the value of constrains'); compare(x_al)disp('the opt point');x=x_al;fmin=obj(X);>> clear>> x_al=[0,0];>> [x,fmin,k]=almain(x_al)上机大作业3：1、>> clear alln=3; c=[-3,-1,-3]'; A=[2,1,1;1,2,3;2,2,1;-1,0,0;0,-1,0;0,0,-1];b=[2,5,6,0,0,0]'; cvx_beginvariable x(n)minimize( c'*x)subject toA*x<=bcvx_endCalling SDPT3 4.0: 6 variables, 3 equality constraints------------------------------------------------------------num. of constraints = 3dim. of linear var = 6*******************************************************************SDPT3: Infeasible path-following algorithms*******************************************************************version predcorr gam expon scale_dataNT 1 0.000 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime-------------------------------------------------------------------0|0.000|0.000|1.1e+01|5.1e+00|6.0e+02|-7.000000e+01 0.000000e+00| 0:0:00| chol 1 11|0.912|1.000|9.4e-01|4.6e-02|6.5e+01|-5.606627e+00 -2.967567e+01| 0:0:01| chol 1 12|1.000|1.000|1.3e-07|4.6e-03|8.5e+00|-2.723981e+00 -1.113509e+01| 0:0:01| chol 1 13|1.000|0.961|2.3e-08|6.2e-04|1.8e+00|-4.348354e+00 -6.122853e+00| 0:0:01| chol 1 14|0.881|1.000|2.2e-08|4.6e-05|3.7e-01|-5.255152e+00 -5.622375e+00| 0:0:01| chol 1 15|0.995|0.962|1.6e-09|6.2e-06|1.5e-02|-5.394782e+00 -5.409213e+00| 0:0:01| chol 1 16|0.989|0.989|2.7e-10|5.2e-07|1.7e-04|-5.399940e+00 -5.400100e+00| 0:0:01| chol 1 17|0.989|0.989|5.3e-11|5.8e-09|1.8e-06|-5.399999e+00 -5.400001e+00| 0:0:01| chol 1 18|1.000|0.994|2.8e-13|4.3e-11|2.7e-08|-5.400000e+00 -5.400000e+00| 0:0:01| stop: max(relative gap, infeasibilities) < 1.49e-08-------------------------------------------------------------------number of iterations = 8primal objective value = -5.39999999e+00dual objective value = -5.40000002e+00gap := trace(XZ) = 2.66e-08relative gap = 2.26e-09actual relative gap = 2.21e-09rel. primal infeas (scaled problem) = 2.77e-13rel. dual " " " = 4.31e-11rel. primal infeas (unscaled problem) = 0.00e+00rel. dual " " " = 0.00e+00norm(X), norm(y), norm(Z) = 4.3e+00, 1.3e+00, 1.9e+00norm(A), norm(b), norm(C) = 6.7e+00, 9.1e+00, 5.4e+00Total CPU time (secs) = 0.71CPU time per iteration = 0.09termination code = 0DIMACS: 3.6e-13 0.0e+00 5.8e-11 0.0e+00 2.2e-09 2.3e-09-------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval): -5.42、>> clear alln=2; c=[-2,-4]'; G=[0.5,0;0,1]; A=[1,1;-1,0;0,-1]; b=[1,0,0]'; cvx_beginvariable x(n)minimize( x'*G*x+c'*x)subject toA*x<=bcvx_endCalling SDPT3 4.0: 7 variables, 3 equality constraintsFor improved efficiency, SDPT3 is solving the dual problem.------------------------------------------------------------num. of constraints = 3dim. of socp var = 4, num. of socp blk = 1dim. of linear var = 3******************************************************************* SDPT3: Infeasible path-following algorithms*******************************************************************version predcorr gam expon scale_dataNT 1 0.000 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime-------------------------------------------------------------------0|0.000|0.000|8.0e-01|6.5e+00|3.1e+02| 1.000000e+01 0.000000e+00| 0:0:00| chol 1 1 1|1.000|0.987|4.3e-07|1.5e-01|1.6e+01| 9.043148e+00 -2.714056e-01| 0:0:00| chol 1 1 2|1.000|1.000|2.6e-07|7.6e-03|1.4e+00| 1.234938e+00 -5.011630e-02| 0:0:00| chol 1 1 3|1.000|1.000|2.4e-07|7.6e-04|3.0e-01| 4.166959e-01 1.181563e-01| 0:0:00| chol 1 1 4|0.892|0.877|6.4e-08|1.6e-04|5.2e-02| 2.773022e-01 2.265122e-01| 0:0:00| chol 1 1 5|1.000|1.000|1.0e-08|7.6e-06|1.5e-02| 2.579468e-01 2.427203e-01| 0:0:00| chol 1 1 6|0.905|0.904|3.1e-09|1.4e-06|2.3e-03| 2.511936e-01 2.488619e-01| 0:0:00| chol 1 1 7|1.000|1.000|6.1e-09|7.7e-08|6.6e-04| 2.503336e-01 2.496718e-01| 0:0:00| chol 1 1 8|0.903|0.903|1.8e-09|1.5e-08|1.0e-04| 2.500507e-01 2.499497e-01| 0:0:00| chol 1 19|1.000|1.000|4.9e-10|3.5e-10|2.9e-05| 2.500143e-01 2.499857e-01| 0:0:00| chol 1 1 10|0.904|0.904|4.7e-11|1.3e-10|4.4e-06| 2.500022e-01 2.499978e-01| 0:0:00| chol 2 2 11|1.000|1.000|2.3e-12|9.4e-12|1.2e-06| 2.500006e-01 2.499994e-01| 0:0:00| chol 2 2 12|1.000|1.000|4.7e-13|1.0e-12|1.8e-07| 2.500001e-01 2.499999e-01| 0:0:00| chol 2 2 13|1.000|1.000|2.0e-12|1.0e-12|4.2e-08| 2.500000e-01 2.500000e-01| 0:0:00| chol 2 2 14|1.000|1.000|2.6e-12|1.0e-12|7.3e-09| 2.500000e-01 2.500000e-01| 0:0:00|stop: max(relative gap, infeasibilities) < 1.49e-08-------------------------------------------------------------------number of iterations = 14primal objective value = 2.50000004e-01dual objective value = 2.49999996e-01gap := trace(XZ) = 7.29e-09relative gap = 4.86e-09actual relative gap = 4.86e-09rel. primal infeas (scaled problem) = 2.63e-12rel. dual " " " = 1.00e-12rel. primal infeas (unscaled problem) = 0.00e+00rel. dual " " " = 0.00e+00norm(X), norm(y), norm(Z) = 3.2e+00, 1.5e+00, 1.9e+00norm(A), norm(b), norm(C) = 3.9e+00, 4.2e+00, 2.6e+00Total CPU time (secs) = 0.36CPU time per iteration = 0.03termination code = 0DIMACS: 3.7e-12 0.0e+00 1.3e-12 0.0e+00 4.9e-09 4.9e-09------------------------------------------------------------------------------------------------------------------------------- Status: SolvedOptimal value (cvx_optval): -3。
教学大纲-大连理工大学教务处
目录《机械设计基础A》 (1)《机械设计基础B》 (8)《**模型设计概论》 (15)阅后删除：请以学部下设学院为单位将全部课程编辑在同一个文档内《机械设计基础A》教学大纲（学分4 学时64）一、课程说明(200字以内，简单说明本课程的地位及教学内容等，阅后删除红色字体）本课程是工科近机械类（包括机械类某些专业）和非机械类专业大类课程之一，是工科学生学习和掌握各种类型的机械中常用机构和通用机械零件的基本知识和基本设计方法的技术基础课。
该课程也是工科学生将来学习专业机械设备课程的理论基础。
本课程在教学内容方面着重基本知识、基本理论和基本设计方法的讲解；在培养实践能力方面着重设计构思和基本设计技能的基本训练。
二、课程目标（对应毕业要求：1-○1、1-○2、1-○3）1. 学习机械工程基础知识和基本理论知识，掌握常用机构的结构、特性等基本知识，了解各种机械的传动原理，具有分析、选用和设计机械设备中基本机构的能力（对应毕业要求：1-○1）；2. 通用机械零件的设计原理、方法和机械设计等的一般规律，具有设计机械传动装置和简单机械的能力（对应毕业要求：1-○1）；3. 掌握基本的机械设计创新方法，培养学生追求创新的态度和意识（对应毕业要求：1-○1）；4. 培养学生树立正确的设计思想，了解机械设计过程中国家有关的经济、环境、法律、安全、健康、伦理等政策和制约因素（对应毕业要求：1-○1）；5. 培养学生的工程实践学习能力，使学生掌握典型零件的实验方法，获得实验技能的基本训练，具有运用标准、规范、手册、图册和查阅有关技术资料的能力（对应毕业要求：1-○1）；6. 了解机械设计的前沿和新发展动向（对应毕业要求：1-○1）。
三、教学内容、基本要求与学时分配序号教学内容教学要求学时教学方式对应课程目标1 一、基本概念1. 研究的对象、内容；2. 机械设计的基本要求和一般设计过程。
1. 了解本课程研究的对象、内容2. 了解机械设计的基本要求、一般设计过程。
(整理)大连理工大学--优化作业----程序.
1.1程序（Java）public class Wolfe_Powell {public static double getFx ( double[] x ) {double x1= x[0]; double x2 = x[1];double Fx= 100 * (x2-x1*x1)* (x2-x1*x1) + (1-x1)* (1-x1) ;return Fx;}public static double[] getDeltFx ( double[] x ) {double x1= x[0]; double x2 = x[1];double [] deltFx = new double[2];deltFx [0] = -400*(x2 - x1* x1) *x1- 2*(1- x1) ;deltFx [1] = 200*(x2- x1 * x1) ;return deltFx ;}public static double getDeltFx_Sk ( double[] deltFx , double[] Sk ) { double a = 0 ;for ( int i = 0 ; i < Sk.length ; i++ ) {a = a + deltFx[ i ] * Sk[ i ] ;}return a ;}public static double getL ( double[] x, double[] s ) {double x1= x[0]; double x2 = x[1];double c1 =0.1 , c2 =0.5 ,a =0 , b=1e8 ,L= 1;double Fx0 , Fx1 ，deltFx1_Sk ,deltFx0_Sk ,temp ,temp2;double[] deltFx0 , deltFx1 ;Fx0 = getFx(x) ;deltFx0 = getDeltFx (x) ;deltFx0_Sk = getDeltFx_Sk( deltFx0 , s) ;temp = c2 * getDeltFx_Sk( deltFx0 , s) ;for ( int i=0;i< 1e8 ; i++){temp2 = -c1 * L * deltFx0_Sk ;x[0] = x1 + L *s[0] ;x[1] = x2 + L *s[1] ;Fx1 = getFx(x) ;deltFx1 = getDeltFx (x) ;deltFx1_Sk = getDeltFx_Sk (deltFx1 , s) ;if( (Fx0 - Fx1 ) >= temp2 && deltFx1_Sk >= temp){break ;}else if( (Fx0 - Fx1 ) < temp2 ){b = L ;L = (L +a) /2 ;}else if ( deltFx1_Sk < temp ) {a = L ;L = ( L + b ) / 2 >= 2*L ? (2*L):( L + b ) / 2;}}System.out.println(" L= " + L);System.out.println(" 计算次数" + i );return L ;}public static void main(String[] args) {Wolfe_Powell temp =new Wolfe_Powell();double [] X = { -1 ,1 } ; double [] sk = { 1 ,1} ; temp.getL( X ,sk) ;}}1.2实验结果步长L = 0.00390625 x =[-0.9992 , 1.0324] 计算次数82.1程序（Java）public class GongE {public static double getFx ( double[] x ) {double x1= x[0];double x2 = x[1];double Fx= x1*x1 - 2*x1*x2 + 2*x2*x2 +x3*x3 - x2*x3 +2 * x1 +3*x2 -x3 ; return Fx;}public static double[] getDeltFx ( double[] x ) {double x1= x[0];double x2 = x[1];double [] deltFx = new double[x.length];deltFx [0] = 2*x1 - 2*x2+2 ;deltFx [1] = -2*x1 +4*x2 - x3 +3;deltFx [2] = 2*x3 -x2 -1 ;return deltFx ;}public static double[] getX ( double[] x ) {double[] g0,g1;double[] s0= new double[x.length];double[] s1=new double[x.length];double g0_L,g1_L ,L ,temp;double[] x0 =x ;int k =0 ;g0 = getDeltFx ( x0 ) ;for ( int j = 0 ; j < x.length ; j++ ) {s0[ j ] = -g0[ j ] ;}for (int i = 0 ;i<2; i ++,k++){g0 = getDeltFx ( x0 ) ;g0_L = getDeltFx_Sk ( s0 , s0 ) ;L =getL(x0,s0); // 例题一中的方法取得步长Lfor(int j=0;j<x.length ; j++){x0[j]= x0[j]+ s0[j]*L ;}g1 = getDeltFx(x0) ;g1_L = getDeltFx_Sk ( g1 , g1 );if ( Math.sqrt( g1_L )<= 1e-2 ) {break ;}else{temp = g1_L/ g0_L ;for(int j=0;j<x.length ; j++){s0[j] = -g1[j] + temp * s0[j];}} }return x0;}public static void main(String[] args) {GongE temp =new GongE();double [] x = { 1,1 } ;double[] result = temp.getX(x) ;for ( int i = 0 ; i < x.length ; i++ ) {System.out.println ( "result[" + i + "]=" + result[ i ] ) ;} } }2.2实验结果最优点x*=[-4,-3,-1] 最优解f*=-83.1公用程序（Java）public static double getFx ( double[] x ) { //取得Fx 值double x1= x[0]; double x2 = x[1];double Fx = x1 + 2 * x2 * x2 + Math.exp ( x1 * x1 + x2 * x2 ) ;return Fx ;}public static double[] getDeltFx ( double[] x ) { //取得Fx 的梯度值double x1= x[0]; double x2 = x[1];double[] deltFx = new double[ 2 ] ;deltFx[ 0 ] = 1 + 2 * x1 * Math.exp ( x1 * x1 + x2 * x2 ) ;deltFx[ 1 ] = 4 * x2 + 2 * x2 * Math.exp ( x1 * x1 + x2 * x2 ) ;return deltFx ;}3.2.1最速下降法程序（Java）public class FastWay {public static double[] getX ( double[] x ) {double [] deltF0 = new double[2]; double L =0;for ( int i = 0 ; i < 1e1 ; i++ ) {deltF0 = getDeltFx(x);for(int j=0 ;j <deltF0.length ;j++){ //取得负梯度deltF0[j] = - deltF0[j];}L = getL ( x , deltF0 ) ; // 调用习题1的不精确搜索取得步长Lif ( Math.sqrt ( getDeltFx_Sk ( deltF0 , deltF0 ) ) <= 1e-3 ) {System.out.println ( "最终计算次数" + i ) ;System.out.println("x1=" + x[0]+" x2=" + x[1]);break ;}x[0] = x[0]+ L * deltF0[ 0 ] ; x[1]= x[1]+ L * deltF0[ 1 ] ;}return x;}public static void main ( String[] args ) {FastWay temp = new FastWay () ;double[] x0 = { 2 , 2} ; temp.getX(x0) ;}3.2.2最速下降法结果最优点X*=[-0.4194 0] 最优解f*=0.7729 计算次数count=10 3.3.1牛顿法程序（Java）public static double[] getDeltFx ( double[] x ) {double x1 = x[ 0 ] ; double x2 = x[ 1 ] ;double[] one = new double[ 2 ] ;double exp =Math.exp( Math.pow(x1,2)+Math.pow(x2,2)) ;one[ 0 ] = 1+ 2*x1*exp ; one[ 1 ] = 4* x2 +2*x2*exp ;double[][] two = new double[2][2] ;two[0][0] = 2*exp *(1+2*Math.pow(x1,2)) ;two[1][1] = 2*exp *(1+2*Math.pow(x2,2)) +4 ;double[] deltFx = new double[ 2 ] ;for (int i = 0 ; i < 2 ; i++ ) {deltFx[0] = one[ 0 ]/two[0][0] ;deltFx[1] = one[ 1 ]/two[1][1] ;}return deltFx;}public static void main ( String[] args ) {double[] x = { 1 , 0} ;double[] DeltFx = new double [2] ;for(int i =0 ;i <1e3;i++){DeltFx = getDeltFx(x);x[0] = x[0]- DeltFx[0];x[1] = x[1]- DeltFx[1];if( Math.sqrt( getDeltFx_Sk(DeltFx,DeltFx ) ) <= 1e-4){System.out.println("计算次数为" + i);break ;}}System.out.println(" x1= " +x[0] +" x2= " + x[1] +"\n") ;System.out.println(" Fx= " +getFx(x)) ;}3.3.2牛顿法结果最优点X*=[ -0.4194 , 0] 最优解f*= 0.7729 计算次数count=5 3.4.1 BFGS法程序（matlab）function [x,val,k] = bfgs(fun,gfun,x0)maxk=1000; sigma=0.4; rho=0.55 ; epsion=1e-5;k=0 ; n =length(x0); Bk=eye(n); %Bk=feval('Hess',x0);while (k<maxk)gk=feval(gfun,x0);if(norm(gk)<epsion),break;end;dk=-Bk\gk;m=0;mk=0;while(m<20)newf=feval(fun,x0+rho^m*dk)oldf=feval(fun,x0)if(newf<oldf+sigma*rho^m*gk'*dk)mk=m;break;endm=m+1;endx=x0+rho^mk*dk;sk=x-x0;yk=feval(gfun,x)-gk;if(yk'*sk>0)Bk=Bk-(Bk*sk*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk);end;k=k+1; x0=x;endval=feval(fun,x0);3.4.2 BFGS法结果最优点X*=[-0.4194 0] 最优解f*=0.7729 计算次数count=44.1 有效集法（matlab）4.1.1 主程序function[x , Lagrange , exitflag , output]= TwoProg (H,c,Ae,be,Ai,bi,x0)n=length(x0); x=x0; ni=length(bi); ne=length(be); Lagrange =zeros(ne+ni,1); index=ones(ni,1); for(i=1:ni)if(Ai(i,:)*x>bi(i)+1e-9),index(i)=0;endend%算法主程序k=0;while(k<=1e4)%求解子问题Temp=[];if(ne>0),Temp=Ae ; endfor(j=1:ni)if(index(j)>0),Temp=[Temp;Ai(j,:)];endendgk=H*x+c;[m1,n1]=size(Temp);[dk,Lagrange ]=SubPro (H,gk , Temp,zeros(m1,1));if(norm(dk)<= 1.0e-6)y=0.0;if(length(Lagrange )>ne)[y,jk]=min(Lagrange (ne+1:length(Lagrange )));endif(y>=0)exitflag=0;elseexitflag=1;for(i=1:ni)if(index(i)&(ne+sum(index(1:i)))==jk)index(i)=0;break;endendendk=k+1;elseexitflag=1;%求步长alpha=1.0;tm=1.0;for(i=1:ni)if((index(i)==0)&(Ai(i,:)*dk<0))tm1=(bi(i)-Ai(i,:)*x)/(Ai(i,:)*dk);if(tm1<tm)tm=tm1;ti=i;endendendalpha=min(alpha,tm);x=x+alpha*dk;if(tm<1),index(ti)=1;endendif(exitflag==0),break;endk=k+1;endoutput.fval=0.5*x'*H*x+c'*x;output.iter=k;4.1.2 目标函数function f=fun(x)x1=x(1); x2=x(2); f=eval ('x1+2*x2^2+exp(x1^2+x2^2)');4.1.3 子问题函数function[x, Lagrange ]= SubPro (H ,c, Ae, be)[m,n]=size(Ae);ginvH=pinv(H);if(m>0)rb=Ae*ginvH*c+be;Lagrange =pinv(Ae*ginvH*Ae')*rb;x=ginvH*(Ae'*Lagrange -c);elsex=-ginvH*c;Lagrange =0;end4.1.4 运行函数H=[2 -2;-2 4];c=[-2 -6]';Ae=[ ];be=[ ];Ai=[1 -2;-0.5 -0.5;1 0;0 1];bi=[-2 -1 0 0]';x0=[0 1 ]';[x,lambda,exitflag,output]=qpact(H,c,Ae,be,Ai,bi,x0)4.2 有效集法结果内部点初始点x0=[0 0] 最优点X*=[0.8 1.2] 最优解f*=-7.2 迭代次数=10 边界点初始点x0=[1 1] 最优点X*=[0.8 1.2] 最优解f*=-7.2 迭代次数=2 检验点初始点x0=[0 1] 最优点X*=[0.8 1.2] 最优解f*=-7.2 迭代次数=75.1 乘子法程序（matlab）5.1.1 chengZi程序---乘子法主程序function[x,mu,Lagrange ,output]=chengZi(fun,hf,gf,dfun,dhf,dgf,x0)sigma=2.0;count=0;innerCount=0;eta=2.0;θ=0.8;%PHR算法中的实参数θx=x0;he=feval(hf,x);gi=feval(gf,x);n=length(x);l=length(he);m=length(gi);%选取乘子向量的初始值mu=0.1*ones(l,1);Lagrange =0.1*ones(m,1);btak=10;btaold=10;%用来检验终止条件的两个值while(btak>1e-6&count<1e3 )%调用BFGS算法程序求解无约束子问题[x,ival,ik]=bfgs('Lagr','LagrTiDu',x0,fun,hf,gf,dfun,dhf,dgf,mu,Lagrange ,sigma);innerCount=innerCount+ik;he=feval(hf,x);gi=feval(gf,x);btak=0.0;for(i=1:l),btak=btak+he(i)^2; endfor(i=1:m)temp=min(gi(i),Lagrange (i)/sigma);btak=btak+temp^2;endbtak=sqrt(btak);if btak>1e-6if(count>=2&btak>θ*btaold)sigma=eta*sigma;end%更新乘子向量for(i=1:l),mu(i)=mu(i)-sigma*he(i);endfor(i=1:m)Lagrange (i)=max(0.0,Lagrange (i)-sigma*gi(i));endendcount=count+1;btaold=btak;x0=x;endf=feval(fun,x)output.inner_iter=innerCount;output.iter=count;output.bta=btak;output.fval=f;5.1.2 f1程序---目标函数function f=f1(x)f=4*x(1)-x(2)^2-12;5.1.3 h1程序---等式约束function he=h1(x)he=25-x(1)^2-x(2)^2;5.1.4 g1程序---不等式约束function gi=g1(x)gi=10*x(1)-x(1)^2+10*x(2)-x(2)^2-34;5.1.5 df1程序---目标函数的梯度文件function g=df1(x)g=[4 ,-2.0*x(2)]';5.1.6 dhe程序---等式约束（向量）函数的Jacobi矩阵（转置）function dhe=dh1(x)dhe=[-2*x(1),-2.0*x(2)]';5.1.7 dgi程序---不等式约束（向量）函数的Jacobi矩阵（转置）function dgi=dg1(x)dgi=[10-2*x(1),10-2*x(2);0,1;1,0]';5.1.8 LagrTiDu程序---增广拉格朗日函数的梯度程序function result=LagrTiDu(x,fun,hf,gf,dfun,dhf,dgf,mu,Lagrange ,sigma) result=feval(dfun,x);he=feval(hf,x);gi=feval(gf,x);dhe=feval(dhf,x);dgi=feval(dgf,x);l=length(he);m=length(gi);for(i=1:l)result=result+(sigma*he(i)-mu(i))*dhe(:,i);精品文档endfor(i=1:m)result=result+(sigma*gi(i)-Lagrange (i))*dgi(:,i);end5.1.9 Lagr程序---增广拉格朗日函数程序function result=Lagr(x,fun,hf,gf,dfun,dhf,dgf,mu,Lagrange ,sigma)f=feval(fun,x);he=feval(hf,x);gi=feval(gf,x);l=length(he);m=length(gi);result=f;s1=0.0;for(i=1:l)result=result-he(i)*mu(i);s1=s1+he(i)^2;endresult=result+0.5*sigma*s1;s2=0.0;for(i=1:m)s3=max(0.0,Lagrange (i)-sigma*gi(i));s2=s2+s3^2-Lagrange (i)^2;endresult=result+s2/(2.0*sigma);5.2 乘子法结果初始点x0=[0 , 0] 最优点X*=[1.0013，4.8987] 最优解f*= -31.9923 等式乘子向量L hu=1.0156 不等式乘子向量Lg=0.75445精品文档。
优化方法大作业1
优化方法大作业一、Wolfe-Powell法利用MATLAB软件编写，其中初始值x0=-5；其他参数按照已知条件来取。
当b分别等于8、9、10时，均得到如下结果：而当初值x0变化时，则结果变化比较大，如将x0取-6，则计算结果如下：通过比较可以看出，b的取值对计算结果的影响较小，而初始值x0的取值则对结果影响很大。
从中也表明Wolfe-Powell准则的收敛条件比较弱，容易出现当函数还没取极小值而迭代循环已结束的情况。
具体代码见附录1.二、无约束优化1.DFP法精度ε = 0.02，确定λ使用的是一维非精确搜索算法(直接法，Wolfe-Powell准则)。
结果如下图所示：2. BFGS方法同上一种方法，精度ε = 0. 02，确定λ使用的是一维非精确搜索算法(直接法，Wolfe-Powell准则)。
结果如下图所示：比较两种方法，查看每一步的函数值，得出如下结果。
图1：DFP与BFGS法结果对比图通过图1与表1的比较，BFGS比DFP法最初收敛得更快，但是DFP法比BFGS法的最终结果更好。
DFP与BFGS法的代码分别见附录2、3。
三、Rockafellar乘子法文件myfun.m：function y=myfun(x)y=1000-x(1)^2-2*x(2)^2-x(3)^2-x(1)*x(2)-x(1)*x(3);文件mycon.m：function [c,ceq]=mycon(x)c(1)=(-x(1));c(2)=(-x(2));c(3)=(-x(3));ceq(1)=x(1)^2+x(2)^2+x(3)^2-25;ceq(2)=8*x(1)+14*x(2)+7*x(3)-56文件Q3.m：clearclcx0=[2 2 2]';[x,fval,exitflag,output]=fmincon(@myfun,x0,[],[],[],[],[],[],@mycon);附录1：%% Wolfe-Powell 准则方法clear;tic;c1=0.1;c2=0.65;a=0;b=8;syms x;f=x*x-2*x+7; %函数G0=jacobian(f,x);%求梯度x0=-6; %初始取x=-5% 由于只有一个未知数，默认S=1lambda = 1;x=x0;for k=1:1:1000f0 = eval(f); %求出函数值grad1= eval(G0); %求出梯度值figure1(k) = f0; % 用于绘出迭代过程中函数值变化x= x0+lambda;f1 = eval(f); %求出函数值grad2 = eval(G0); %求出梯度值value1 = f0 - f1 + c1* lambda * grad1 ;value2 = grad2 -c2*grad1 ;if value1 >=-1e-6 && value2 >=-1e-6disp('The variable matrix is : ');disp(x);fun=eval(f);fprintf('The minimum value of function is %f \n',fun);break; %如果满足两个条件，则退出循环。
大连理工优化方法大作业MATLAB编程
function [x,dk,k]=fjqx(x,s) flag=0;a=0;b=0;k=0;d=1;while(flag==0)[p,q]=getpq(x,d,s);if (p<0)b=d;d=(d+a)/2;endif(p>=0)&&(q>=0)dk=d;x=x+d*s;flag=1;endk=k+1;if(p>=0)&&(q<0)a=d;d=min{2*d,(d+b)/2};endend%定义求函数值的函数fun，当输入为x0=（x1，x2）时，输出为f function f=fun(x)f=(x(2)-x(1)^2)^2+(1-x(1))^2;function gf=gfun(x)gf=[-4*x(1)*(x(2)-x(1)^2)+2*(x(1)-1),2*(x(2)-x(1)^2)]; function [p,q]=getpq(x,d,s)p=fun(x)-fun(x+d*s)+0.20*d*gfun(x)*s';q=gfun(x+d*s)*s'-0.60*gfun(x)*s';结果：x=[0,1];s=[-1,1];[x,dk,k]=fjqx(x,s)x =-0.0000 1.0000dk =1.1102e-016k =54function f= fun( X )%所求问题目标函数f=X(1)^2-2*X(1)*X(2)+2*X(2)^2+X(3)^2+ X(4)^2-X(2)*X(3)+2*X(1)+3*X(2)-X(3);endfunction g= gfun( X )%所求问题目标函数梯度g=[2*X(1)-2*X(2)+2,-2*X(1)+4*X(2)-X(3)+3,2*X(3)-X(2)-1,2*X(4)];endfunction [ x,val,k ] = frcg( fun,gfun,x0 )%功能：用FR共轭梯度法求无约束问题最小值%输入：x0是初始点，fun和gfun分别是目标函数和梯度%输出：x、val分别是最优点和最优值，k是迭代次数maxk=5000;%最大迭代次数rho=0.5;sigma=0.4;k=0;eps=10e-6;n=length(x0);while(k<maxk)g=feval(gfun,x0);%计算梯度itern=k-(n+1)*floor(k/(n+1));itern=itern+1;%计算搜索方向if(itern==1)d=-g;elsebeta=(g*g')/(g0*g0');d=-g+beta*d0;gd=g'*d;if(gd>=0.0)d=-g;endendif(norm(g)<eps)break;endm=0;mk=0;while(m<20)if(feval(fun,x0+rho^m*d)<feval(fun,x0)+sigma*rho^m*g'*d) mk=m;break;endm=m+1;endx0=x0+rho^mk*d;val=feval(fun,x0);g0=g;d0=d;k=k+1;endx=x0;val=feval(fun,x0);end结果：>> x0=[0,0,0,0];>> [ x,val,k ] = frcg( 'fun','gfun',x0 )x =-4.0000 -3.0000 -1.0000 0val =-8.0000k =21或者function [x,f,k]=second(x)k=0;dk=dfun(x);g0=gfun(x);s=-g0;x=x+dk*s;g1=gfun(x);while(norm(g1)>=0.02)if(k==3)k=0;g0=gfun(x);s=-g0;x=x+dk*s;g1=gfun(x);else if(k<3)u=((norm(g1))^2)/(norm(g0)^2); s=-g1+u*s;k=k+1;g0=g1;dk=dfun(x);x=x+dk*s;g1=gfun(x);endendf=fun(x);endfunction f=fun(x)f=x(1)^2-2*x(1)*x(2)+2*x(2)^2+x(3)^2+x(4)^2-x(2)*x(3)+2*x(1)+3*x(2)-x(3); function gf=gfun(x)gf=[2*x(1)-2*x(2)+2,-2*x(1)+4*x(2)-x(3)+3,2*x(3)-x(2)-1,2*x(4)];function [p,q]=con(x,d)ss=-gfun(x);p=fun(x)-fun(x+d*ss)+0.2*d*gfun(x)*(ss)';q=gfun(x+d*ss)*(ss)'-0.6*gfun(x)*(ss)';function dk=dfun(x)flag=0;a=0;d=1;while(flag==0)[p,q]=con(x,d);if (p<0)b=d;d=(d+a)/2;endif(p>=0)&&(q>=0)dk=d;flag=1;endif(p>=0)&&(q<0)a=d;d=min{2*d,(d+b)/2};endEnd结果：x=[0,0,0,0];>> [x,f,k]=second(x)x =-4.0147 -3.0132 -1.0090 0 f = -7.9999k = 1function [f,x,k]=third_1(x)k=0;g=gfun(x);while(norm(g)>=0.001)s=-g;dk=dfun(x,s);x=x+dk*s;k=k+1;g=gfun(x);f=fun(x);endfunction f=fun(x)f=x(1)+2*x(2)^2+exp(x(1)^2+x(2)^2);function gf=gfun(x)gf=[1+2*x(1)*exp(x(1)^2+x(2)^2),4*x(2)+2*x(2)*(x(1)^2+x(2)^2)];function [j_1,j_2]=con(x,d,s)j_1=fun(x)-fun(x+d*s)+0.1*d*gfun(x)*(s)'; j_2=gfun(x+d*s)*(s)'-0.5*gfun(x)*(s)'; function dk=dfun(x,s)%获取步长flag=0;a=0;d=1;while(flag==0)[p,q]=con(x,d,s);if (p<0)b=d;d=(d+a)/2;endif(p>=0)&&(q>=0)dk=d;flag=1;endif(p>=0)&&(q<0)a=d;d=min{2*d,(d+b)/2};endend结果：x=[0,1];[f,x,k]=third_1(x)f =0.7729x = -0.4196 0.0001k =8（1）程序：function [f,x,k]=third_2(x)k=0;H=inv(ggfun(x));g=gfun(x);while(norm(g)>=0.001)s=(-H*g')';dk=dfun(x,s);x=x+dk*s;k=k+1;g=gfun(x);f=fun(x);endfunction f=fun(x)f=x(1)+2*x(2)^2+exp(x(1)^2+x(2)^2); function gf=gfun(x)gf=[1+2*x(1)*exp(x(1)^2+x(2)^2),4*x(2)+2*x(2)*(x(1)^2+x(2)^2)]; function ggf=ggfun(x)ggf=[(4*x(1)^2+2)*exp(x(1)^2+x(2)^2),4*x(1)*x(2)*exp(x(1)^2+x(2)^2);4*x(1)*x(2)*exp(x(1)^2+x(2)^2),4+(4*x(2)^2+2)*exp(x(1)^2+x(2)^2)]; function [j_1,j_2]=con(x,d,s)j_1=fun(x)-fun(x+d*s)+0.1*d*gfun(x)*(s)';j_2=gfun(x+d*s)*(s)'-0.5*gfun(x)*(s)';function dk=dfun(x,s)% 步长获取flag=0;a=0;d=1;b=10000;while(flag==0)[p,q]=con(x,d,s);if (p<0)b=d;d=(d+a)/2;endif(p>=0)&&(q>=0)dk=d;flag=1;endif(p>=0)&&(q<0)a=d;if 2*d>=(d+b)/2d=(d+b)/2;else d=2*d;endendEnd结果：x=[0,1];[f,x,k]=third_2(x)f =0.7729x = -0.4193 0.0001k =8（2）程序：function [f,x,k]=third_3(x) k=0;X=cell(2);g=cell(2);X{1}=x;H=eye(2);g{1}=gfun(X{1});s=(-H*g{1}')';dk=dfun(X{1},s);X{2}=X{1}+dk*s;g{2}=gfun(X{2});while(norm(g{2})>=0.001)dx=X{2}-X{1};dg=g{2}-g{1};v=dx/(dx*dg')-(H*dg')'/(dg*H*dg'); h1=H*dg'*dg*H/(dg*H*dg');h2=dx'*dx/(dx*dx');h3=dg*H*dg'*v'*v;H=H-h1+h2+h3;k=k+1;X{1}=X{2};g{1}=gfun(X{1});s=(-H*g{1}')';dk=dfun(X{1},s);X{2}=X{1}+dk*s;g{2}=gfun(X{2});norm(g{2});f=fun(x);x=X{2};endfunction f=fun(x)f=x(1)+2*x(2)^2+exp(x(1)^2+x(2)^2);function gf=gfun(x)gf=[1+2*x(1)*exp(x(1)^2+x(2)^2),4*x(2)+2*x(2)*(x(1)^2+x(2)^2)];function ggf=ggfun(x)ggf=[(4*x(1)^2+2)*exp(x(1)^2+x(2)^2),4*x(1)*x(2)*exp(x(1)^2+x(2)^2);4*x(1)*x(2)* exp(x(1)^2+x(2)^2),4+(4*x(2)^2+2)*exp(x(1)^2+x(2)^2);function [p,q]=con(x,d,s)p=fun(x)-fun(x+d*s)+0.1*d*gfun(x)*(s)';q=gfun(x+d*s)*(s)'-0.5*gfun(x)*(s)';function dk=dfun(x,s)flag=0;a=0;d=1;b=10000;while(flag==0)[p,q]=con(x,d,s);if (p<0)b=d;d=(d+a)/2;endif(p>=0)&&(q>=0) dk=d;flag=1;endif(p>=0)&&(q<0)a=d;if 2*d>=(d+b)/2d=(d+b)/2;else d=2*d;endendend结果：x=[0,1];[f,x,k]=third_3(x)f =0.7729x = -0.4195 0.0000 k=6function callqpactH=[2 0; 0 2];c=[-2 -5]';Ae=[ ]; be=[ ];Ai=[1 -2; -1 -2; -1 2;1 0;0 1];bi=[-2 -6 -2 0 0]';x0=[0 0]';[x,lambda,exitflag,output]=qpact(H,c,Ae,be,Ai,bi,x0) function [x,lamk,exitflag,output]=qpact(H,c,Ae,be,Ai,bi,x0) epsilon=1.0e-9; err=1.0e-6;k=0; x=x0; n=length(x); kmax=1.0e3;ne=length(be); ni=length(bi); lamk=zeros(ne+ni,1); index=ones(ni,1);for (i=1:ni)if(Ai(i,:)*x>bi(i)+epsilon), index(i)=0; endendwhile(k<=kmax)Aee=[];if(ne>0), Aee=Ae; endfor(j=1:ni)if(index(j)>0), Aee=[Aee; Ai(j,:)]; end endgk=H*x+c;[m1,n1] = size(Aee);[dk,lamk]=qsubp(H,gk,Aee,zeros(m1,1)); if(norm(dk)<=err)y=0.0;if(length(lamk)>ne)[y,jk]=min(lamk(ne+1:length(lamk))); endif(y>=0)exitflag=0;elseexitflag=1;for(i=1:ni)if(index(i)&(ne+sum(index(1:i)))==jk) index(i)=0; break;endendendk=k+1;elseexitflag=1;alpha=1.0; tm=1.0;for(i=1:ni)if((index(i)==0)&(Ai(i,:)*dk<0)) tm1=(bi(i)-Ai(i,:)*x)/(Ai(i,:)*dk); if(tm1<tm)tm=tm1; ti=i;endendendalpha=min(alpha,tm);x=x+alpha*dk;if(tm<1), index(ti)=1; end endif(exitflag==0), break; endk=k+1;endoutput.fval=0.5*x'*H*x+c'*x; output.iter=k;function [x,lambda]=qsubp(H,c,Ae,be) ginvH=pinv(H);[m,n]=size(Ae);if(m>0)rb=Ae*ginvH*c + be;lambda=pinv(Ae*ginvH*Ae')*rb;x=ginvH*(Ae'*lambda-c);elsex=-ginvH*c;lambda=0;end结果>>callqpactx =1.40001.7000lambda =0.8000exitflag =output =fval: -6.4500iter: 7function [x,mu,lambda,output]=multphr(fun,hf,gf,dfun,dhf,dgf,x0)%功能: 用乘子法解一般约束问题: min f(x), s.t. h(x)=0, g(x).=0%输入: x0是初始点, fun, dfun分别是目标函数及其梯度；% hf, dhf分别是等式约束（向量）函数及其Jacobi矩阵的转置；% gf, dgf分别是不等式约束（向量）函数及其Jacobi矩阵的转置；%输出: x是近似最优点，mu, lambda分别是相应于等式约束和不等式约束的乘子向量; % output是结构变量, 输出近似极小值f, 迭代次数, 内迭代次数等maxk=500;c=2.0;eta=2.0;theta=0.8;k=0;ink=0;epsilon=0.00001;x=x0;he=feval(hf,x);gi=feval(gf,x);n=length(x);l=length(he);m=length(gi);mu=zeros(l,1);lambda=zeros(m,1);btak=10;btaold=10;while(btak>epsilon&&k<maxk)%调用BFGS算法程序求解无约束子问题[x,ival,ik]=bfgs('mpsi','dmpsi',x0,fun,hf,gf,dfun,dhf,dgf,mu,lambda,c);ink=ink+ik;he=feval(hf,x);gi=feval(gf,x);btak=0;for i=1:lbtak=btak+he(i)^2;end%更新乘子向量for i=1:mtemp=min(gi(i),lambda(i)/c);btak=btak+temp^2;endbtak=sqrt(btak);if btak>epsilonif k>=2&&btak>theta*btaoldc=eta*c;endfor i=1:lmu(i)=mu(i)-c*he(i);endfor i=1:mlambda(i)=max(0,lambda(i)-c*gi(i));endk=k+1;btaold=btak;x0=x;endendf=feval(fun,x);output.fval=f;output.iter=k;%增广拉格朗日函数function psi=mpsi(x,fun,hf,gf,dfun,dhf,dgf,mu,lambda,c) f=feval(fun,x);he=feval(hf,x);gi=feval(gf,x);l=length(he);m=length(gi);psi=f;s1=0;for i=1:lpsi=psi-he(i)*mu(i);s1=s1+he(i)^2;endpsi=psi+0.5*c*s1;s2=0;for i=1:ms3=max(0,lambda(i)-c*gi(i));s2=s2+s3^2-lambda(i)^2;endpsi=psi+s2/(2*c);%不等式约束函数文件g1.mfunction gi=g1(x)gi=10*x(1)-x(1)^2+10*x(2)-x(2)^2-34;%目标函数的梯度文件df1.mfunction g=df1(x)g=[4, -2*x(2)]';%等式约束（向量）函数的Jacobi矩阵（转置）文件dh1.m function dhe=dh1(x)dhe=[-2*x(1), -2*x(2)]'%不等式约束（向量）函数的Jacobi矩阵（转置）文件dg1.m function dgi=dg1(x)dgi=[10-2*x(1), 10-2*x(2)]';function [x,val,k]=bfgs(fun,gfun,x0,varargin)maxk=500;rho=0.55;sigma=0.4;epsilon=0.00001;k=0;n=length(x0);Bk=eye(n);while(k<maxk)gk=feval(gfun,x0,varargin{:});if(norm(gk)<epsilon)break;enddk=-Bk\gk;m=0;mk=0;while(m<20)newf=feval(fun,x0+rho^m*dk,varargin{:});oldf=feval(fun,x0,varargin{:});if(newf<oldf+sigma*rho^m*gk'*dk)mk=m;break;endm=m+1;endx=x0+rho^mk*dk;sk=x-x0;yk=feval(gfun,x,varargin{:})-gk;if(yk'*sk>0)Bk=Bk-(Bk*sk*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk);endk=k+1;x0=x;endval=feval(fun,x0,varargin{:});结果x=[2 2]';[x,mu,lambda,output]=multphr('fun','hf','gf1','df','dh','dg',x0) x =1.00134.8987mu =0.7701lambda =0.9434output =fval: -31.9923iter: 4f=[3,1,1];A=[2,1,1;1,-1,-1];b=[2;-1];lb=[0,0,0];x=linprog(f,A,b,zeros(3),[0,0,0]',lb)结果：Optimization terminated.x =0.00000.50000.5000。
大连理工大学c++大作业
大连理工大学C++程序设计大作业班级：111111111姓名：暗暗暗暗暗暗暗暗学号：1111111111邮箱：1111111111111111任课教师：赵国辉上交时间：2013.7.22目录1.第一次上机作业 (3)2.第二次上机作业 (6)3.第三次上机作业 (10)4.第四次上机作业 (15)5.结课大作业 (18)6.课堂总结 (48)1. 第一次上机作业1.1作业要求1.整型和浮点型的二进制表示2.一个整数的二进制（64位）转换成十进制表示。
1.2 核心代码说明整型和浮点型的二进制表示#include <iostream>#include <cstdio>#include <cstring>#include <algorithm>#include <queue>#include <stack>using namespace std;void getint(int x){stack<int>s;while(x){s.push(x&1);x>>=1;}while(!s.empty()){cout<<s.top();s.pop();}} //将整型转换成二进制函数void getfloat(float y){queue<int>q;int x=(int)y;getint(x);y-=x; //将浮点型转换成二进制函数if(!y) return ;putchar('.');while(y){y*=2.0;if(y>=1.0){q.push(1);y-=1.0;}else q.push(0);}while(!q.empty()){cout<<q.front();q.pop();}}int main(){int x;float y;cin>>x;//输入一个整数getint(x);cout<<endl;cin>>y;//输入一个浮点数getfloat(y);cout<<endl;return 0;}一个整数的二进制（64位）转换成十进制表示。
大连理工优化方法-增广拉格朗日方法MATLAB程序
大连理工优化方法-增广拉格朗日方法MATLAB程序上机大作业II定义目标函数funfunction f=fun(x)x1=x(1);x2=x(2);f=4*x1-x2^2-12;定义目标函数梯度函数dfunfunction f=dfun(x)x2=x(2);f=[4;-2*x2];定义等式约束函数hffunction qua=hf(x)qua=25-x(1)^2-x(2)^2;定义等式约束函数梯度函数dhffunction qua=dhf(x)qua=[-2*x(1);-2*x(2)];定义不等式约束函数gfunfunction inq=gfun(x)inq=10*x(1)-x(1)^2+10*x(2)-x(2)^2-34;定义不等式约束梯度数dgffunction inq=dgf(x)inq=[10-2*x(1);10-2*x(2)];定义增广拉格朗日函数mpsifunctionpsi=mpsi(x,fun,hf,gfun,dfun,dhf,dgf,mu,lambda,sigma)f=feval(fun,x);he=feval(hf,x);gi=feval(gfun,x);l=length(he);m=length(gi);psi=f;s1=0;for i=1:lpsi=psi-he(i)*mu(i);s1=s1+he(i)^2;endpsi=psi+0.5*sigma*s1;s2=0.0;for i=1:ms3=max(0.0, lambda(i) - sigma*gi(i));s2=s2+s3^2-lambda(i)^2;endpsi=psi+s2/(2.0*sigma);定义增广拉格朗日函数梯度函数dmpsifunctiondpsi=dmpsi(x,fun,hf,gfun,dfun,dhf,dgf,mu,lambda,sigma) dpsi=feval(dfun,x);he=feval(hf,x);gi=feval(gfun,x);dhe=feval(dhf,x);dgi=feval(dgf,x);l=length(he);m=length(gi);for i=1:ldpsi=dpsi+(sigma*he(i)-mu(i))*dhe(:,i);endfor i=1:mdpsi=dpsi+(sigma*gi(i)-lambda(i))*dgi(:,i);end定义BFGS法函数函数bfgsfunction[x,val,k]=bfgs(mpsi,dmpsi,x0,fun,hf,gfun,dfun,dhf,dgf,mu,lambda ,sigma) maxk=1000;rho=0.5;sigma1=0.4;epsilon1=1e-4;k=0;n=length(x0);Bk=eye(n);while(k<maxk)< p="">gk=feval(dmpsi,x0,fun,hf,gfun,dfun,dhf,dgf,mu,lambda,sigm a);if(norm(gk)<epsilon1)< p="">break;enddk=-Bk\gk;m=0;mk=0;while(m<20)newf=feval(mpsi,x0+rho^m*dk,fun,hf,gfun,dfun,dhf,dgf,mu,l ambda,sigma);oldf=feval(mpsi,x0,fun,hf,gfun,dfun,dhf,dgf,mu,lambda,sigma);if(newf<oldf+sigma1*rho^m*gk'*dk)< p="">mk=m;break;endm=m+1;endx=x0+rho^mk*dk;sk=x-x0;yk=feval(dmpsi,x,fun,hf,gfun,dfun,dhf,dgf,mu,lambda,sigma) -gk;if(yk'*sk>0)Bk=Bk-((Bk*sk)*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk);endk=k+1;x0=x;endval=feval(mpsi,x0,fun,hf,gfun,dfun,dhf,dgf,mu,lambda,sigma );定义增广拉格朗日乘子法函数multphrfunction answer=multphr(fun,hf,gfun,dfun,dhf,dgf,x0)maxk=5000;sigma=2.0;eta=2.0;theta=0.8;k=0;ink=0;epsilon=1e-4;x=x0;he=feval(hf,x);gi=feval(gfun,x);l=length(he);m=length(gi);mu=0.1*ones(l,1);lambda=0.1*ones(m,1);btak=10;btaold=10;while(btak>epsilon&&k<maxk)< p="">[x,v,ik]=bfgs('mpsi','dmpsi',x0,fun,hf,gfun,dfun,dhf,dgf,mu,la mbda,sigma); ink=ink+ik;he=feval(hf,x);gi=feval(gfun,x);btak=0.0;for i=1:lbtak=btak+he(i)^2;endfor i=1:mtemp=min(gi(i),lambda(i)/sigma);btak=btak+temp^2;endbtak=sqrt(btak);if btak>epsilonif(k>=2&&btak > theta*btaold)sigma=eta*sigma;endfor i=1:lmu(i)=mu(i)-sigma*he(i);endfor i=1:mlambda(i)=max(0.0,lambda(i)-sigma*gi(i)); endendk=k+1;btaold=btak;x0=x;endf=feval(fun,x);xfmulambdak运行求解>> x0=[0;0]x0 =>> multphr('fun','hf','gfun','dfun','dhf','dgf',x0) x = 1.001281489564374.89871784708758f =-31.9923105871169mu =1.01559644571312lambda =0.754451167977228k =4</maxk)<></oldf+sigma1*rho^m*gk'*dk)<></epsilon1)<></maxk)<>。
大连理工大学结构优化复习总结
⼤连理⼯⼤学结构优化复习总结结构优化设计－基于结构分析技术，在给定的设计空间实现满⾜使⽤要求且具有最佳性能或最低成本的⼯程结构设计的技术优化设计的三要素:设计变量;约束条件;⽬标函数凸域:基于n维空间的区域s⾥，如果取任意两点x1和x2，连接这两点的线段也属于s，该区域称凸域（=αx1+(1-α)x2 ）凸函数:如果函数f(x)定义在n维空间的凸域s上，⽽且对s中的任意两点x1和x2和任意常数α,0.0<=α<=1.0,有f[αx1+(1- α)x2]<=αf(x1)+(1- α)f(x2),则f(x)称为s上的凸函数严格凸函数:上式⼩于严格成⽴凸规划:如果可⾏域是凸域，⽬标函数是凸函数，这样构成的数学规划问题为凸规划问题。
准则设计法:依靠⼯程经验;效率⾼;缺乏严格数学基础最优准则法基于库塔克（K-T）条件:需构造迭代求解算法;通⽤性不强数学规划⽅法:有严格的数学基础，有较好的通⽤性，计算效率要考虑。
结构优化问题的求解布骤I. 建⽴优化模型。
给定初始设计⽅案。
II. 结构分析(有限元）III.优化（收敛性）检验。
满⾜则结束程序，否则继续IVIV. 灵敏度分析V. 求解优化问题，修改结构模型，返回II。
优化求解的两⼤类⽅法：准则法；数学规划法准则设计⽅法：⽤优化准则代替原来的优化问题同步失效准则设计的评价：{优点：简单、⽅便，特别是独⽴约束个数n=m时；⼯程实⽤；适合于构件设计。
缺点：只能处理简单构件设计；缩⼩了设计空间，不能保证最优解；若n < m ，可能⽆解；当n > m时，确定哪些破坏模式应同时发⽣⽐较困难。
改进：为了弥补等式约束代替不等式约束的缺陷，引⼊松弛因⼦ψiσi (X ) =ψiσip , 0 ≤ψi ≤1, i =1,2,......n启发：⽤准则代替原来的优化问题，准则法的基本思想；如果将桁架的每根杆看作⼀种可能的破坏模式，桁架看作⼀个元件。
可以得到满应⼒准则满应⼒⽅法的缺点：完全⽆视重量会漏掉最轻设计；中间点⼀般是不可⾏设计，对⼯程实际不利。
大连理工大学机械设计大作业
目录一、设计任务书及原始数据 (2)二、根据已知条件计算传动件的作用力 (3)2.1计算齿轮处转矩T、圆周力F t 、径向力F r及轴向力F a .. 3 2.2计算链轮作用在轴上的压力 (3)2.3计算支座反力 (4)三、初选轴的材料，确定材料的机械性能 (4)四、进行轴的结构设计 (5)4.1确定最小直径 (5)4.2设计其余各轴段的直径和长度，且初选轴承型号 (5)4.3选择连接形式与设计细部结构 (6)五．轴的疲劳强度校核 (6)5.1轴的受力图 (6)5.2绘制弯矩图 (7)5.3绘制转矩图 (8)5.4确定危险截面 (9)5.5计算当量应力，校核轴的疲劳强度 (9)六、选择轴承型号，计算轴承寿命 (10)6.1计算轴承所受支反力 (10)6.2计算轴承寿命 (11)七、键连接的计算 (11)八、轴系部件的结构装配图 (12)一、设计任务书及原始数据题目二：二级展开式斜齿圆柱齿轮减速器输出轴组合结构设计表1 设计方案及原始数据二、根据已知条件计算传动件的作用力2.1计算齿轮处转矩T、圆周力F t、径向力F r及轴向力F a已知：轴输入功率P=4.3kW，转速n=130r/(min)。
(1)齿轮上的力转矩计算公式：T=9.550×106P/n将数据代入公式中，得：T=315885(N·mm)圆周力计算公式：Ft=2T/，==416(mm) (认为是法面模数)将转矩T带入其中，得：Ft=1519(N)径向力计算公式：Fr =Ft×tanα/cos，=将圆周力Ft 带入其中，得：Fr=558(N)轴向力计算公式：Fa = Ft×tan将圆周力Ft 带入其中，得：Fa=216(N)2.2计算链轮作用在轴上的压力链轮的分度园直径链速v=链的圆周力F=链轮作用在轴上的压力2.3计算支座反力1、计算垂直面（XOZ）支反力根据受力分析图，我们可以利用垂直面力矩平衡原理（ΣMY=0）得出求解A点垂直面支反力Rz1：R z1= Ft1- Rz2ΣM a= R Z2× l AC- F t1× l AB=0即ΣMa =1519 ×135-RZ2× 215=0RZ2=954 NRz1=565 N2、计算水平面（XOY）支反力根据受力分析图，我们可以利用水平面力矩平衡原理（ΣMZ=0）得出求解A点水平面支反力Ry1的计算公式：R y1= FQ- Ry2–FrΣM Z= R y2× l AC+ F r× l AB +F a×r- F Q× l ADΣM Z=R y2× 215+558×135+216×135-×315=0Ry2=4437NRy1=- 4437 –558=-1635 N三、初选轴的材料，确定材料的机械性能初选材料及机械性能见表四、进行轴的结构设计4.1确定最小直径按照扭转强度条件计算轴的最小值d min。
大连理工大学 秋季优化方法大作业
m=m+1; end x0=x0+rho^mk*d; val=feval(fun,x0); g0=g; d0=d; k=k+1; end x=x0; val=feval(fun,x);
//f(x)//
function f=fun(x) x1=[1 0]*x; x2=[0 1]*x; f=(1-x1)^2+100*(x2-x1^2)^2; //梯度函数// function g=gfun(x) x1=[1 0]*x; x2=[0 1]*x; g=[-2*(1-x1)-400*x1*(x2-x1^2); 200*(x2-x1^2)]; //运行过程// >> x0=[0 0]'
k=k+1; btaold=btak; x0=x; end f=feval(fun,x); output.fval=f; output.iter=k; output.inner_iter=ink; output.bta=btak;
//增广拉格朗日函数//
function psi=mpsi(x,fun,hf,gf,dfun,dhf,dgf,mu,lambda,sigma) f=feval(fun,x); he=feval(hf,x); gi=feval(gf,x); l=length(he); m=length(gi); psi=f; s1=0.0; for(i=1:l) psi=psi-he(i)*mu(i); s1=s1+he(i)^2; end psi=psi+0.5*sigma*s1; s2=0.0; for(i=1:m) s3=max(0.0, lambda(i) - sigma*gi(i)); s2=s2+s3^2-lambda(i)^2; end psi=psi+s2/(2.0*sigma); //增广拉格朗日函数// function dpsi=dmpsi(x,fun,hf,gf,dfun,dhf,dgf,mu,lambda,sigma) dpsi=feval(dfun,x); he=feval(hf,x); gi=feval(gf,x); dhe=feval(dhf,x); dgi=feval(dgf,x); l=length(he); m=length(gi); for(i=1:l) dpsi=dpsi+(sigma*he(i)-mu(i))*dhe(:,i); end for(i=1:m) dpsi=dpsi+(sigma*gi(i)-lambda(i))*dgi(:,i); end //f(x)// function f=f1(x) f=4*x(1)-x(2)^2-12; //等式约束// function he=h1(x) he=25-x(1)^2-x(2)^2;
大连理工大学优化方法上机作业
优化方法上机大作业学院：电子信息与电气工程学部姓名：学号：指导老师：上机大作业（一）%目标函数function f=fun(x)f=100*(x(2)-x(1)^2)^2+(1-x(1))^2;end%目标函数梯度function gf=gfun(x)gf=[-400*x(1)*(x(2)-x(1)^2)-2*(1-x(1));200*(x(2)-x(1)^2)];End%目标函数Hess矩阵function He=Hess(x)He=[1200*x(1)^2-400*x(2)+2,-400*x(1);-400*x(1), 200;];end%线搜索步长function mk=armijo(xk,dk)beta=; sigma=;m=0; maxm=20;while (m<=maxm)if(fun(xk+beta^m*dk)<=fun(xk)+sigma*beta^m*gfun(xk)'*dk)mk=m; break;endm=m+1;endalpha=beta^mknewxk=xk+alpha*dkfk=fun(xk)newfk=fun(newxk)%最速下降法function [k,x,val]=grad(fun,gfun,x0,epsilon)%功能：梯度法求解无约束优化问题：minf(x)%输入：fun，gfun分别是目标函数及其梯度，x0是初始点，% epsilon为容许误差%输出：k是迭代次数，x，val分别是近似最优点和最优值maxk=5000; %最大迭代次数beta=; sigma=;k=0;while(k<maxk)gk=feval(gfun,x0); %计算梯度dk=-gk; %计算搜索方向if(norm(gk)<epsilon), break;end%检验终止准则m=0;mk=0;while(m<20) %用Armijo搜索步长if(feval(fun,x0+beta^m*dk)<=feval(fun,x0)+sigma*beta^m*gk'*dk) mk=m;break;endm=m+1;endx0=x0+beta^mk*dk;k=k+1;endx=x0;val=feval(fun,x0);>> x0=[0;0];>> [k,x,val]=grad('fun','gfun',x0,1e-4)迭代次数：k =1033x =val =%牛顿法x0=[0;0];ep=1e-4;maxk=10;k=0;while(k<maxk)gk=gfun(x0);if(norm(gk)<ep)x=x0miny=fun(x)k0=kbreak;elseH=inv(Hess(x0));x0=x0-H*gk;k=k+1;endendx =miny =迭代次数k0 =2%BFGS方法function [k,x,val]=bfgs(fun,gfun,x0,varargin)%功能：梯度法求解无约束优化问题：minf(x)%输入：fun，gfun分别是目标函数及其梯度，x0是初始点，% epsilon为容许误差%输出：k是迭代次数，x，val分别是近似最优点和最优值N=1000;epsilon=1e-4;beta=;sigma=;n=length(x0);Bk=eye(n);k=0;while(k<N)gk=feval(gfun,x0,varargin{:});if(norm(gk)<epsilon), break;enddk=-Bk\gk;m=0;mk=0;while(m<20)newf=feval(fun,x0+beta^m*dk,varargin{:});oldf=feval(fun,x0,varargin{:});if(newf<=oldf+sigma*beta^m*gk'*dk)mk=m;break;endm=m+1;endx=x0+beta^mk*dk;sk=x-x0;yk=feval(gfun,x,varargin{:})-gk;if(yk'*sk>0)Bk=Bk-(Bk*sk*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk);endk=k+1;x0=x;endval=feval(fun,x0,varargin{:});>> x0=[0;0];>> [k,x,val]=bfgs('fun','gfun',x0)k =20x =val =%共轭梯度法function [k,x,val]=frcg(fun,gfun,x0,epsilon,N)if nargin<5,N=1000;endif nargin<4, epsilon=1e-4;endbeta=;sigma=;n=length(x0);k=0;while(k<N)gk=feval(gfun,x0);itern=k-(n+1)*floor(k/(n+1));itern=itern+1;if(itern==1)dk=-gk;elsebetak=(gk'*gk)/(g0'*g0);dk=-gk+betak*d0; gd=gk'*dk;if(gd>=0),dk=-gk;endendif(norm(gk)<epsilon),break;endm=0;mk=0;while(m<20)if(feval(fun,x0+beta^m*dk)<=feval(fun,x0)+sigma*beta^m*gk'*dk) mk=m;break;endm=m+1;endx=x0+beta^m*dk;g0=gk; d0=dk;x0=x;k=k+1;endval=feval(fun,x);>> x0=[0;0];[k,x,val]=frcg('fun','gfun',x0,1e-4,1000)k =122x =val =上机大作业（二）%目标函数function f_x=fun(x)f_x=4*x(1)-x(2)^2-12;%等式约束条件function he=hf(x)he=25-x(1)^2-x(2)^2;end%不等式约束条件function gi_x=gi(x,i)switch icase 1gi_x=10*x(1)-x(1)^2+10*x(2)-x(2)^2-34;case 2gi_x=x(1);case 3gi_x=x(2);otherwiseend%求目标函数的梯度function L_grad=grad(x,lambda,cigma)d_f=[4;2*x(2)];d_g(:,1)=[-2*x(1);-2*x(2)];d_g(:,2)=[10-2*x(1);10-2*x(2)];d_g(:,3)=[1;0];d_g(:,4)=[0;1];L_grad=d_f+(lambda(1)+cigma*hf(x))*d_g(:,1);for i=1:3if lambda(i+1)+cigma*gi(x,i)<0L_grad=L_grad+(lambda(i+1)+cigma*gi(x,i))*d_g(:,i+1);continueendend%增广拉格朗日函数function LA=lag(x,lambda,cee)LA=fun(x)+lambda(1)*hf(x)+*cee*hf(x)^2;for i=1:3LA=LA+1/(2*cee)*(min(0,lambda(i+1)+cee*gi(x,i))^2-lambda(i+1)^2);endfunction xk=BFGS(x0,eps,lambda,cigma)gk=grad(x0,lambda,cigma);res_B=norm(gk);k_B=0;a_=1e-4;rho=;c=1e-4;length_x=length(x0);I=eye(length_x);Hk=I;while res_B>eps&&k_B<=10000dk=-Hk*gk;m=0;while m<=5000if lag(x0+a_*rho^m*dk,lambda,cigma)-lag(x0,lambda,cigma)<=c*a_*rho^m*gk'*dk mk=m;break;endm=m+1;endak=a_*rho^mk;xk=x0+ak*dk;delta=xk-x0;y=grad(xk,lambda,cigma)-gk;Hk=(I-(delta*y')/(delta'*y))*Hk*(I-(y*delta')/(delta'*y))+(delta*delta')/(delta'*y); k_B=k_B+1;x0=xk;gk=y+gk;res_B=norm(gk);end%增广拉格朗日法function val_min=ALM(x0,eps)lambda=zeros(4,1);cigma=5;alpha=10;k=1;res=[abs(hf(x0)),0,0,0];for i=1:3res(1,i+1)=norm(min(gi(x0,i),-lambda(i+1)/cigma));endres=max(res);while res>eps&&k<1000xk=BFGS(x0,eps,lambda,cigma);lambda(1)=lambda(1)+cigma*hf(xk);for i=1:3lambda(i+1)=lambda(i+1)+min(0,lambda(i+1)+gi(x0,1));endk=k+1;cigma=alpha*cigma;x0=xk;res=[norm(hf(x0)),0,0,0];for i=1:3res(1,i+1)=norm(min(gi(x0,i),-lambda(i+1)/cigma)); endres=max(res);endval_min=fun(xk);fprintf('k=%d\n',k);fprintf('fmin=%.4f\n',val_min);fprintf('x=[%.4f;%.4f]\n',xk(1),xk(2));>> x0=[0;0];>> val_min=ALM(x0,1e-4)k=10fmin=x=[;]val_min =上机大作业（三）A=[1 1;-1 0;0 -1];n=2;b=[1;0;0];G=[ 0;0 2];c=[2 4];cvx_solver sdpt3cvx_beginvariable x(n)minimize (x'*G*x-c*x)subject toA*x<=bcvx_enddisp(x)Status: SolvedOptimal value (cvx_optval):A=[2 1 1;1 2 3;2 2 1;-1 0 0;0 -1 0;0 0 -1]; n=3;b=[2;5;6;0;0;0];C=[-3 -1 -3];cvx_solver sdpt3cvx_beginvariable x(n)minimize (C*x)subject toA*x<=bcvx_enddisp(x)Status: SolvedOptimal value (cvx_optval):。
大连理工优化方法大作业MATLAB编程
fun ctio n [x,dk,k]=fjqx(x,s) flag=0;a=0;b=0;k=0;d=1;while (flag==0)[p,q]=getpq(x,d,s);if (P<0)b=d;d=(d+a)/2;endif(p>=0) &&( q>=0)dk=d;x=x+d*s;flag=1;endk=k+1;if (p>=0)&&(q<0)a=d;d=min{2*d,(d+b)/2};endend%定义求函数值的函数 fun ，当输入为 x0= (x1 ， x2 )时，输出为 f function f=fun(x)f=(x(2)-x(1)A2)A2+(1-x(1)F2;function gf=gfun(x)gf=[-4*x(1)*(x (2) -x(1)A2)+2*(x(1)-1),2*(x(2)-x(1)A2)];function [p,q]=getpq(x,d,s)p=fun(x)-fun(x+d*s)+0.20*d*gfun(x)*s';q=gfun(x+d*s)*s'-0.60*gfun(x)*s';结果：x=[0,1];s=[-1,1];[x,dk,k]=fjqx(x,s)x =-0.0000 1.0000dk =1.1102e-016k =54取初始= (0.0. 0,0)r^'l用兵柜梯皮法求解下面无约東优化问题：min f (x) = x孑—2x^X2 十2x孑 + x孑H-爲—X2天3 十 2xj + 3|X2 —*3,其中步长g的选取可利用习題1戎精确一维披索.注:通过比习题验证共範梯度法求辉门无二次西数极小点至多需要“次迭代.fun ctio n f= fun( X )%所求问题目标函数f=X(1)A2-2*X(1)*X (2)+2*X(2)A2+X(3)A2+ X(4) A2-X( 2)*X(3)+2*X(1)+3*X(2)-X(3);end function g= gfun( X )%所求问题目标函数梯度g=[2*X(1)-2*X(2)+2,-2*X(1)+4*X(2)-X(3)+3,2*X (3) -X (2)-1,2*X(4)];end function [ x,val,k ] = frcg( fun,gfun,xO )%功能：用FR共轭梯度法求无约束问题最小值%输入：x0是初始点，fun和gfun分别是目标函数和梯度%输出：x、val分别是最优点和最优值，k是迭代次数maxk=5000; %最大迭代次数rho=0.5;sigma=0.4;k=0;eps=10e-6;n=length(x0);while (k<maxk)g=feval(gfun,x0); % 计算梯度 itern=k-(n+1)*floor(k/(n+1));itern=itern+1;%计算搜索方向if (itern==1)d=-g;elsebeta=(g*g')/(g0*g0');d=-g+beta*d0;gd=g'*d;if (gd>=0.0)d=-g;endendif (norm(g)<eps)break ;endm=0;mk=0;while (m<20)if(feval(fu n,xO+rhoAm*d)<feval(fu n,xO)+sigma*rhoAm*g'*d) mk=m; break ;endm=m+1;endx0=x0+rho A mk*d;val=feval(fun,x0);g0=g;d0=d;k=k+1;endx=x0;val=feval(fun,x0);end结果：>> x0=[0,0,0,0];>> [ x,val,k ] = frcg( 'fun','gfun',x0 ) x =-4.0000 -3.0000 -1.0000 0val =-8.0000k =或者function [x,f,k]=second(x)k=0;dk=dfun(x);g0=gfun(x);s=-g0;x=x+dk*s;g1=gfun(x);while (norm(g1)>=0.02)if (k==3)k=0;g0=gfun(x);s=-g0;x=x+dk*s;g1=gfun(x);else if (k<3)u=(( norm(g1))A2)/( norm(gO)A2); s=-g1+u*s;k=k+1;g0=g1;dk=dfun(x);x=x+dk*s;g1=gfun(x);endendf=fun(x);endfunction f=fun(x)f=x(1F2-2*x(1)*x (2)+2*x (2)A2+x(3)A2+x(4)A2-x (2) *x (3)+2*x(1)+3*x(2)-x(3); function gf=gfun(x)gf=[2*x(1)-2*x(2)+2,-2*x(1)+4*x(2)-x(3)+3,2*x(3)-x(2)-1,2*x(4)];function [p,q]=con(x,d)ss=-gfun(x);p=fun(x)-fun(x+d*ss)+0.2*d*gfun(x)*(ss)';q=gfun(x+d*ss)*(ss)'-0.6*gfun(x)*(ss)';function dk=dfun(x)flag=0;a=0;d=1;while (flag==0)[p,q]=con(x,d);if (p<0)b=d;d=(d+a)/2;endif (p>=0)&&(q>=0)dk=d;flag=1;endif (p>=0)&&(q<0)a=d;d=min{2*d,(d+b)/2};endEnd结果： x=[0,0,0,0];>> [x,f,k]=second(x)x =-4.0147 -3.0132-1.0090 0 f = -7.9999k = 1取初始点3 = （0」）二考虑下面无约東优化问题：min f（x）二冷 + 2x2 + exp（xf + 天孑），其中歩长Qk的选取可別用习题1或精确一维搜索•搜索方向为一HNW ♦取垃=b•取皿=R2f防）］"9耳丈啟为BFG5公式亠通过此习题体会上述三种算法的收敛速度.fun ctio n [f,x,k]=third_1(x) k=0;g=gfu n(x);while (norm(g)>=0.001) s=-g;dk=dfu n( x,s);x=x+dk*s;k=k+1;g=gfu n(x);f=fun( x);endfun ctio n f=fun(x)f=x(1)+2*x(2)A2+exp(x(1)A2+x(2)A2);fun ctio n gf=gfu n(x)gf=[1+2*x(1)*exp(x(1)A2+x(2)A2),4*x(2)+2*x(2)*(x(1)A2+x(2)A2)]; function[j_1,j_2]=con(x,d,s)j_1=fun(x)-fun(x+d*s)+0.1*d*gfun(x)*(s)'; j_2=gfun(x+d*s)*(s)'-0.5*gfun(x)*(s)'; function dk=dfun(x,s) % 获取步长 flag=0;a=0;d=1;while (flag==0)[p,q]=con(x,d,s);if (p<0)b=d;d=(d+a)/2;endif (p>=0)&&(q>=0)dk=d;flag=1;endif (p>=0)&&(q<0)a=d;d=min{2*d,(d+b)/2}; end结果：x=[0,1];[f,x,k]=third_1(x)f =0.7729x = -0.4196 0.0001k =8(1 ) 程序：function [f,x,k]=third_2(x)k=0;H=inv(ggfun(x));g=gfun(x);while (norm(g)>=0.001)s=(-H*g')';dk=dfun(x,s);x=x+dk*s;k=k+1;g=gfun(x);f=fun(x);endfunction f=fun(x)f=x(1)+2*x(2)A2+exp(x(1F2+x(2)A2);function gf=gfun(x) gf=[1+2*x(1)*exp(x(1F2+x(2)A2),4*x(2)+2*x(2)*(x(1F2+x(2)A2)]; function ggf=ggfun(x)ggf=[(4*x(1)A2+2)*exp(x(1)A2+x (2) A2),4*x(1)*x (2) *exp(x(1)A2+x(2)A2);4*x(1)*x(2)*exp(x(1)A2+x(2)A2),4+(4*x(2)A2+2)*exp(x(1)A2+x(2)A2)];function [j_1,j_2]=con(x,d,s)j_1=fun(x)-fun(x+d*s)+0.1*d*gfun(x)*(s)';j_2=gfun(x+d*s)*(s)'-0.5*gfun(x)*(s)'; function dk=dfun(x,s) % 步长获取flag=0;a=0;d=1;b=10000;while (flag==0)[p,q]=con(x,d,s);if (p<0)b=d;d=(d+a)/2;endif(p>=0)&&(q>=0)dk=d;flag=1;endif (p>=0)&&(q<0)a=d;if 2*d>=(d+b)/2d=(d+b)/2;endendEnd结果：x=[0,1];[f,x,k]=third_2(x)f =0.7729x = -0.4193 0.0001k =8(2) 程序：function [f,x,k]=third_3(x) k=0;X=cell(2);g=cell(2);X{1}=x;H=eye(2);g{1}=gfun(X{1});s=(-H*g{1}')';dk=dfun(X{1},s);X{2}=X{1}+dk*s;g{2}=gfun(X{2});while (norm(g{2})>=0.001)dg=g{2}-g{1};v=dx/(dx*dg')-(H*dg')'/(dg*H*dg');h1=H*dg'*dg*H/(dg*H*dg');h2=dx'*dx/(dx*dx');h3=dg*H*dg'*v'*v;H=H-h1+h2+h3;k=k+1;X{1}=X{2};g{1}=gfun(X{1});s=(-H*g{1}')';dk=dfun(X{1},s);X{2}=X{1}+dk*s;g{2}=gfun(X{2});norm(g{2});f=fun(x);x=X{2};endfunction f=fun(x)f=x(1)+2*x(2)A2+exp(x(1F2+x(2)A2);function gf=gfun(x)gf=[1+2*x(1)*exp(x(1)A2+x(2)A2),4*x(2)+2*x(2)*(x(1)A2+x(2)A2)];function ggf=ggfun(x)ggf=[(4*x(1)A2+2)*exp(x(1)A2+x(2)A2),4*x(1)*x(2)*exp(x(1)A2+x(2)A2);4*x(1)*x(2)* exp(x(1)A2+x(2)A2),4+(4*x(2)A2+2)*exp(x(1)A2+x(2)A2);function [p,q]=con(x,d,s)p=fun(x)-fun(x+d*s)+0.1*d*gfun(x)*(s)';q=gfun(x+d*s)*(s)'-0.5*gfun(x)*(s)';function dk=dfun(x,s)flag=0;a=0;d=1;b=10000;while (flag==0)[p,q]=con(x,d,s);if (p<0)b=d;d=(d+a)/2;if (p>=0)&&(q>=0)dk=d;flag=1;endif (p>=0)&&(q<0)a=d;if 2*d>=(d+b)/2d=(d+b)/2;else d=2*d;endendend结果：x=[0,1];[f,x,k]=third_3(x)f =0.7729x = -0.41950.0000 k=6*U 用有效集法求解下面勺勺二次规划问题:(XI 一 I)2 + (x 2 一 2.5)2 X1 - 2X2 + 2 > 0-Xi — 2>(2 + 6 > 0-Xi + 2X2 + 2 > 0xi,x 2 > 0function callqpactH=[2 0; 0 2];c=[-2 -5]';Ae=[ ]; be=[];Ai=[1 -2; -1 -2; -1 2;1 0;0 1];bi=[-2 -6 -2 0 0]';x0=[0 0]';[x,lambda,exitflag,output]=qpact(H,c,Ae,be,Ai,bi,xO)fun ctio n [x,lamk,exitflag,output]=qpact(H,c,Ae,be,Ai,bi,x0) epsilo n=1.0e-9; err=1.0e-6;k=0; x=x0; n=len gth(x); kmax=1.0e3;n e=le ngth(be); ni=le ngth(bi); lamk=zeros( ne+n i,1); in dex=ones(n i,1);for (i=1:ni)if(Ai(i,:)*x>bi(i)+epsil on), i ndex(i)=0; end while (k<=kmax)mmSi.Aee=[];if (ne>0), Aee=Ae; endfor (j=1:ni)if (index(j)>0), Aee=[Aee; Ai(j,:)]; end endgk=H*x+c;[m1,n1] = size(Aee);[dk,lamk]=qsubp(H,gk,Aee,zeros(m1,1)); if (norm(dk)<=err)y=0.0;if (length(lamk)>ne)[y,jk]=min(lamk(ne+1:length(lamk))); endif (y>=0)exitflag=0;elseexitflag=1;for (i=1:ni)if (index(i)&(ne+sum(index(1:i)))==jk) index(i)=0; break ;endendk=k+1;elseexitflag=1;alpha=1.0; tm=1.0;for (i=1:ni)if ((index(i)==0)&(Ai(i,:)*dk<0))tm1=(bi(i)-Ai(i,:)*x)/(Ai(i,:)*dk);if (tm1<tm)tm=tm1; ti=i;endendendalpha=min(alpha,tm);x=x+alpha*dk;if (tm<1), index(ti)=1; endendif (exitflag==0), break ; endk=k+1;endoutput.fval=0.5*x'*H*x+c'*x;output.iter=k;function [x,lambda]=qsubp(H,c,Ae,be) ginvH=pinv(H); [m,n]=size(Ae);if (m>0)rb=Ae*ginvH*c + be;lambda=pinv(Ae*ginvH*Ae')*rb; x=ginvH*(Ae'*lambda-c);elsex=-ginvH*c;lambda=0;end结果>>callqpactx =1.40001.7000lambda =0.8000exitflag =output =fval: -6.4500iter: 7function [x,mu,lambda,output]=multphr(fu n, hf,gf,dfu n, dhf,dgf,xO)%功能：用乘子法解一般约束问题：min f(x), s.t. h(x)=0, g(x).=0%输入：x0是初始点，fun, dfun分别是目标函数及其梯度；% hf, dhf分别是等式约束(向量)函数及其 Jacobi矩阵的转置；% gf, dgf分别是不等式约束(向量)函数及其 Jacobi矩阵的转置；%输出：x是近似最优点，mu, lambda分别是相应于等式约束和不等式约束的乘子向量% output是结构变量，输出近似极小值f,迭代次数，内迭代次数等maxk=500;c=2.0;eta=2.0;theta=0.8;k=0;i nk=0;epsilo n=0.00001;x=xO;he=feval(hf,x);gi=feval(gf,x);n=len gth(x);l=le ngth(he);m=le ngth(gi);mu=zeros(l,1);lambda=zeros(m,1);btak=10;btaold=10;while (btak>epsilon&&k<maxk)%调用BFGS算法程序求解无约束子问题[x,ival,ik]=bfgs( 'mpsi' ,'dmpsi' ,x0,fun,hf,gf,dfun,dhf,dgf,mu,lambda,c);ink=ink+ik;he=feval(hf,x);gi=feval(gf,x);btak=0;for i=1:lbtak=btak+he(y2;end% 更新乘子向量for i=1:mtemp=min(gi(i),lambda(i)/c);btak=btak+temp A2;endbtak=sqrt(btak);if btak>epsilonif k>=2&&btak>theta*btaoldc=eta*c;endfor i=1:lmu(i)=mu(i)-c*he(i);endlambda(i)=max(0,lambda(i)-c*gi(i));endk=k+1;btaold=btak;x0=x;endendf=feval(fun,x);output.fval=f;output.iter=k;%增广拉格朗日函数function psi=mpsi(x,fun,hf,gf,dfun,dhf,dgf,mu,lambda,c) f=feval(fun,x);he=feval(hf,x);gi=feval(gf,x);l=length(he);m=length(gi);psi=f;s1=0;for i=1:lpsi=psi-he(i)*mu(i);s仁 s1+he(y2;psi=psi+0.5*c*s1;s2=0;for i=1:ms3=max(0,lambda(i)-c*gi(i));s2=s2+s3A2-lambda(i)A2;endpsi=psi+s2/(2*c);% 不等式约束函数文件 g1.mfunction gi=g1(x)gi=10*x(1)-x(1)A2+10*x(2)-x(2)A2-34;% 目标函数的梯度文件df1.mfunction g=df1(x)g=[4, -2*x(2)]';% 等式约束(向量)函数的Jacobi 矩阵(转置)文件 dh1.m function dhe=dh1(x)dhe=[-2*x(1), -2*x(2)]'% 不等式约束(向量)函数的Jacobi 矩阵(转置)文件 dg1.m function dgi=dg1(x)dgi=[10-2*x(1), 10-2*x(2)]';function [x,val,k]=bfgs(fun,gfun,x0,varargin) maxk=500; rho=0.55;sigma=0.4;epsilon=0.00001;k=0;n=length(x0);Bk=eye(n);while (k<maxk)gk=feval(gfun,x0,varargin{:});if (norm(gk)<epsilon)break ;enddk=-Bk\gk;m=0;mk=0;while (m<20)n ewf=feval(fu n, x0+rho A m*dk,vararg in {:});oldf=feval(fun,x0,varargin{:});if(newf<oldf+sigma*rhoAm*gk'*dk) mk=m;break ;endm=m+1;endx=x0+rhoAmk*dk;sk=x-x0;yk=feval(gfun,x,varargin{:})-gk;if (yk'*sk>0)Bk=Bk-(Bk*sk*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk);endk=k+1;x0=x;endval=feval(fun,x0,varargin{:});结果x=[2 2]';[x,mu,lambda,output]=multphr( 'fun' ,'hf' ,'gf1' ,'df' ,'dh' ,'dg' ,x0) x =1.00134.8987mu =0.7701lambda =0.9434output =fval: -31.9923iter: 4利用序列二次规划方法求解习题5中的约束优化问题:min 4xi 一好一 12s.t. 25 - x? —x孑=Q10x一召 + 10旳-xj - 34 > 0 X1,X2 > 0tf=[3,1,1]；A=[2,1,1;1,-1,-1];b=[2;-1];lb=[0,0,0]； x=li nprog(f,A,b,zeros(3),[0,0,0]',lb)结果：Optimization terminated.0.00000.50000.5000。
最优化方法大作业
发动机空燃比控制器引言：我主要从事自动化相关研究。
这里介绍我曾经接触过的发动机空燃比控制器设计中的优化问题。
发动机空燃比控制器设计中的最优化问题AFR =afm m && (1)空燃比由方程（1）定义，在发动机运行过程中如果控制AFR 稳定在14.7可以获得最好的动力性能和排放性能。
如果假设进入气缸的空气流量am &可以由相关单元检测得到，则可以通过控制进入气缸的燃油流量f m &来实现空燃比的精确控制。
由于实际发动机的燃油喷嘴并不是直接对气缸喷燃油，而是通过进气歧管喷燃油，这么做会在进气歧管壁上液化形成油膜，因此不仅是喷嘴喷出的未液化部分燃油会进入气缸，油膜蒸发部分燃油也会进入气缸，如方程（2）。
这样如何更好的喷射燃油成为了一个问题。
1110101122211ττττ⎡⎤⎡⎤-⎢⎥⎢⎥⎡⎤⎡⎤⎢⎥⎢⎥=+⎢⎥⎢⎥⎢⎥⎢⎥-⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎢⎥-⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎡⎤⎢⎥⎣⎦⎢⎥⎣⎦f ff v X x x u x x X x y =x && (2)其中12、,==ff fv x m x m &&=f y m &,=fi u m &这里面，表示油膜蒸发量ff m &、fvm &表示为液化部分燃油、fim &表示喷嘴喷射的燃油，在τf 、τv 、X 都已知的情况下，由现代控制理论知识，根据系统的增广状态空间模型方程（3）00000011011011114.70ττττ⎡⎤⎡⎤-⎡⎤⎢⎥⎢⎥⎡⎤⎡⎤⎢⎥=-+-⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎢⎥⎢⎥⎣⎦⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎡⎤=⎢⎥⎣⎦⎣⎦ff v v a X X u +q q m y q x x x &&& (3)其中()014.7⎰taq =y -m&。
由极点配置方法，只要设计控制器方程（4），就可以使得y 无差的跟踪阶跃输入，那么y 也能较好的跟踪AFR *am /&。
优化方法课程大作业
优化方法课程上机大作业学部：电子信息与电气工程学部专业：生物医学工程班级：电信硕1303学号：21309210姓名：史益新大连理工大学Dalian University of Technology解：（1）MATLAB代码如下：clc;clear all;close all;[x,y]=meshgrid(-2:0.1:2,-1:0.1:3);z=(y-x.^2).^2+(1-x).^2;mesh(x,y,z)hold on;xk=[0 1]';epsilon=1e-5;plot(xk(1),xk(2),'ro');text(xk(1),xk(2),'start point');hold on;[ x,val,k ]=Newton('fun','gfun','Hess',xk,epsilon)plot(x(1),x(2),'ro');text(x(1),x(2),'end point');function [ x,val,k ] = Newton( fun,gfun,Hess,xk,epsilon ) k=0;while(1)gk=feval(gfun,xk);hk=feval(Hess,xk);sk=-inv(hk)*gk;if(norm(gk)<epsilon)break;endxk=xk+sk;k=k+1;endx=xk;val=feval(fun,x);endfunction f = fun(x)f=(x(2)-x(1)^2)^2+(1-x(1))^2;endfunction g=gfun(x)g=[-4*x(1)*(x(2)-x(1)^2)+2*(1-x(1)),2*(x(2)-x(1)^2)]'; endfunction He = Hess( x )n=length(x);He=zeros(n,n);He=[12*x(1)^2-4*x(2)+2,-4*x(1);-4*x(1), 2 ];end（2）代码运行结果如下：解：（1）MATLAB代码如下：clc;clear all;close all;x0=[0 0 0 0]';epsilon=1e-5;[ x,val,k ]=Frcg('fun','gfun',x0,epsilon)function [ x,val,k ] = Frcg( fun,gfun,x0,epsilon )rho=0.6;sigma=0.5;k=0;n=length(x0);while(1)g=feval(gfun,x0);itern=k-(n+1)*floor(k/(n+1));itern=itern+1;if(itern==1)d=-g;elsebeta=(g'*g)/(g0'*g0);d=-g+beta*d0;gd=g'*d;if(gd>=0.0)d=-g;endendif(norm(g)<epsilon)break;endm=0;mk=0;while(m<20)if(feval(fun,x0+rho^m*d)<feval(fun,x0)+sigma*rho^m*g'*d) mk=m;break;endm=m+1;endx0=x0+rho^mk*d;val=feval(fun,x0);g0=g;d0=d;k=k+1;endx=x0;val=feval(fun,x);function f = fun( x )f=x(1)^2-2*x(1)*x(2)+2*x(2)^2+x(3)^2+x(4)^2-x(2)*x(3)+2*x(1)+3*x(2)-x(3); endfunction g = gfun( x )g=[2*x(1)-2*x(2)+2,-2*x(1)+4*x(2)-x(3)+3,2*x(3)-x(2)-1,2*x(4)]';end（2）代码运行结果如下：解：（1）MATLAB代码如下：clc;clear all;close all;[x,y]=meshgrid(-2:0.1:2,-1:0.1:3);z=5*(y-x.^2).^2+(x-1).^2;mesh(x,y,z)hold on;x0=[2 0]';epsilon=1e-5;plot(x0(1),x0(2),'ro');text(x0(1),x0(2),'start point');hold on;[ x1,val1,k1 ]=grad('fun','gfun',x0,epsilon)plot(x1(1),x1(2),'ro');text(x1(1),x1(2),'end point1');hold on;[ x2,val2,k2 ]=znNewton('fun','gfun','Hess',x0,epsilon) plot(x2(1),x2(2),'bo');text(x2(1),x2(2),'end point2');hold on;[ x3,val3,k3 ]=BFGS('fun','gfun',x0,epsilon)plot(x3(1),x3(2),'go');text(x3(1),x3(2),'end point3');hold on;function [ x,val,k ] = grad( fun,gfun,x0,epsilon )rho=0.5;sigma=0.4;k=0;while(1)g=feval(gfun,x0);d=-g;if(norm(d)<epsilon)break;endm=0;mk=0;while(m<20)if(feval(fun,x0+rho^m*d)<feval(fun,x0)+sigma*rho^m*g'*d) mk=m;break;endm=m+1;endx0=x0+rho^mk*d;k=k+1;endx=x0;val=feval(fun,x0);endfunction [ x,val,k ] = znNewton( fun,gfun,Hess,x0,epsilon )rho=0.5;sigma=0.4;k=0;while(1)gk=feval(gfun,x0);hk=feval(Hess,x0);dk=-inv(hk)*gk;if(norm(gk)<epsilon)break;endm=0;mk=0;while(m<20)if(feval(fun,x0+rho^m*dk)<feval(fun,x0)+sigma*rho^m*gk'*dk) mk=m;break;endm=m+1;endx0=x0+rho^mk*dk;k=k+1;endx=x0;val=feval(fun,x0);endfunction [ x,val,k ] = BFGS( fun,gfun,x0,epsilon )rho=0.5;sigma=0.4;k=0;n=length(x0);bk=eye(n);while(1)gk=feval(gfun,x0);if(norm(gk)<epsilon)break;enddk=-inv(bk)*gk;m=0;mk=0;while(m<20)newf=feval(fun,x0+rho^m*dk);oldf=feval(fun,x0);if(newf<oldf+sigma*rho^m*gk'*dk)mk=m;break;endm=m+1;endx=x0+rho^mk*dk;sk=x-x0;yk=feval(gfun,x)-gk;if(yk'*sk>0)bk=bk-(bk*sk*sk'*bk)/(sk'*bk*sk)+(yk*yk')/(yk'*sk);endk=k+1;x0=x;endval=feval(fun,x0);endfunction f = fun(x)f=5*(x(2)-x(1)^2)^2+(x(1)-1)^2;endfunction g=gfun(x)g=[-20*x(1)*(x(2)-x(1)^2)+2*(x(1)-1),10*(x(2)-x(1)^2)]'; endfunction He = Hess( x )n=length(x);He=zeros(n,n);He=[60*x(1)^2-20*x(2)+2,-20*x(1);-20*x(1), 10 ];end（2）代码运行结果如下：解：（1）MATLAB程序如下：clc;close all;clear all;x0=[1,0]';epsilon=1e-5;[ x,mu,lambda,output ] = multphr( 'f1','h1','g1','df1','dh1','dg1',x0,epsilon )function [ x,mu,lambda,output ] = multphr( fun,hf,gf,dfun,dhf,dgf,x0,epsilon )%MULTPHR Summary of this function goes here% Detailed explanation goes heresigma=2.0;eta=2.0;theta=0.8;k=0;ink=0;x=x0;he=feval(hf,x);gi=feval(gf,x);n=length(x);l=length(he);m=length(gi);%initial of multi-vectormu=0.1*ones(l,1);lambda=0.1*ones(m,1);btak=10;btaold=10;while(btak>epsilon)[x,ival,ik]=BFGS('mpsi','dmpsi',x0,epsilon,fun,hf,gf,dfun,dhf,dgf,mu,lambda,sigma);ink=ink+ik;he=feval(hf,x);gi=feval(gf,x);btak=0;for(i=1:l)btak=btak+he(i)^2;endfor(i=1:m)temp=min(gi(i),lambda(i)/sigma);btak=btak+temp^2;endbtak=sqrt(btak);if(btak>epsilon)if(k>=2 & btak>theta*btaold)sigma=eta*sigma;endfor(i=1:l)mu(i)=mu(i)-sigma*he(i);endfor(i=1:m)lambda(i)=max(0,lambda(i)-sigma*gi(i));endendk=k+1;btaold=btak;x0=x;endf=feval(fun,x);output.fval=f;output.iter=k;output.inner_iter=ink;output.bta=btak;endfunction [ x,val,k ] = BFGS( fun,gfun,x0,varargin )rho=0.5;epsilon=1e-5;sigma=0.4;k=0;n=length(x0);bk=eye(n);while(1)gk=feval(gfun,x0,varargin{:});if(norm(gk)<epsilon)break;enddk=-inv(bk)*gk;m=0;mk=0;while(m<20)newf=feval(fun,x0+rho^m*dk,varargin{:});oldf=feval(fun,x0,varargin{:});if(newf<oldf+sigma*rho^m*gk'*dk)mk=m;break;endm=m+1;endx=x0+rho^mk*dk;sk=x-x0;yk=feval(gfun,x,varargin{:})-gk;if(yk'*sk>0)bk=bk-(bk*sk*sk'*bk)/(sk'*bk*sk)+(yk*yk')/(yk'*sk);endk=k+1;x0=x;endval=feval(fun,x0,varargin{:});endfunction psi = mpsi( x,epsilon,fun,hf,gf,dfun,dhf,dgf,mu,lambda,sigma ) %MPSI Summary of this function goes here% Detailed explanation goes heref=feval(fun,x);he=feval(hf,x);gi=feval(gf,x);l=length(he);m=length(gi);psi=f;s1=0;for(i=1:l)psi=psi-he(i)*mu(i);s1=s1+he(i)^2;endpsi=psi+0.5*sigma*s1;s2=0;for(i=1:m)s3=max(0,lambda(i)-sigma*gi(i));s2=s2+s3^2-lambda(i)^2;endpsi=psi+s2/(2*sigma);endfunction dpsi = dmpsi( x,epsilon,fun,hf,gf,dfun,dhf,dgf,mu,lambda,sigma ) %DMPSI Summary of this function goes here% Detailed explanation goes heredpsi=feval(dfun,x);he=feval(hf,x);gi=feval(gf,x);dhe=feval(dhf,x);dgi=feval(dgf,x);l=length(he);m=length(gi);for(i=1:l)dpsi=dpsi+(sigma*he(i)-mu(i))*dhe(:,i);endfor(i=1:m)dpsi=dpsi+(sigma*gi(i)-lambda(i))*dgi(:,i);endendfunction f = f1( x )f=4*x(1)-x(2)^2-12;endfunction gi = g1( x )gi=10*x(1)-x(1)^2+10*x(2)-x(2)^2-34;%unequation constrainendfunction he = h1( x )he=25-x(1)^2-x(2)^2;%equation constrainendfunction g = df1( x )g=[4,-2*x(2)]';endfunction dgi = dg1( x )dgi=[-2*x(1)+10,-2*x(2)+10]';endfunction dhe = dh1( x )dhe=[-2*x(1),-2*x(2)]';end（2）代码运行结果如下：解：（1）MATLAB代码如下：clc;close all;clear all;H=[2 0;0 2];c=[-2 -5]';Ae=[];be=[];Ai=[1 -2;-1 -2;-1 2;1 0;0 1];bi=[-2 -6 -2 0 0]';x0=[0 0]';epsilon=1e-9;[ x,lamk,exitflag,output ] = qpact( H,c,Ae,be,Ai,bi,x0,epsilon )function [ x,lamk,exitflag,output ] = qpact( H,c,Ae,be,Ai,bi,x0,epsilon ) %QPACT Summary of this function goes here% Detailed explanation goes hereerr=1e-6;k=0;x=x0;n=length(x);kmax=1e3;ne=length(be);ni=length(bi);lamk=zeros(ne+ni,1);index=ones(ni,1);for(i=1:ni)if(Ai(i,:)*x>bi(i)+epsilon)index(i)=0;endendwhile(k<=kmax)Aee=[];if(ne>0)Aee=Ae;endfor(j=1:ni)if(index(j)>0)Aee=[Aee;Ai(j,:)];endendgk=H*x+c;[m1,n1]=size(Aee);[dk,lamk]=qsubp(H,gk,Aee,zeros(m1,1));if(norm(dk)<=err)y=0;if(length(lamk)>ne)[y,jk]=min(lamk(ne+1:length(lamk)));endif(y>=0)exitflag=0;elseexitflag=1;for(i=1:ni)if(index(i)&(ne+sum(index(1:i)))==jk)index(i)=0;break;endendendk=k+1;elseexitflag=1;alpha=1;tm=1;for(i=1:ni)if((index(i)==0)&(Ai(i,:)*dk<0))tm1=(bi(i)-Ai(i,:)*x)/(Ai(i,:)*dk);if(tm1<tm)tm=tm1;ti=i;endendendalpha=min(alpha,tm);x=x+alpha*dk;if(tm<1)index(ti)=1;endendif(exitflag==0)break;end %updata the setk=k+1;endoutput.fval=0.5*x'*H*x+c'*x;output.iter=k;endfunction [ x,lambda ] = qsubp( H,c,Ae,be )%QSUBP Summary of this function goes here % Detailed explanation goes hereginvH=pinv(H);[m,n]=size(Ae);if(m>0)rb=Ae*ginvH*c+be;lambda=pinv(Ae*ginvH*Ae')*rb;x=ginvH*(Ae'*lambda-c);elsex=-ginvH*c;lambda=0;endend（2）代码运行结果如下：解：（1）MATLAB代码如下：clc;close all;clear all;x0=[0.5 0.2]';mu0=[ ]';lam0=[0 0 0 0]';epsilon=1e-6;[ x,mu,lam,val,k ] = sqpm( x0,mu0,lam0,epsilon)function [ x,mu,lam,val,k ] = sqpm( x0,mu0,lam0,epsilon ) %SQPM Summary of this function goes here% Detailed explanation goes heren=length(x0);l=length(mu0);m=length(lam0);rho=0.5;eta=0.1;B0=eye(n);x=x0;mu=mu0;lam=lam0;Bk=B0;sigma=0.8;[hk,gk]=cons(x);dfk=df1(x);[Ae,Ai]=dcons(x);Ak=[Ae;Ai];k=0;while(1)[dk,mu,lam]=qpsubp(dfk,Bk,Ae,hk,Ai,gk,epsilon);mp1=norm(hk,1)+norm(max(-gk,0),1);if (norm(dk,1)<epsilon) & (mp1<1e-5)break;enddeta=0.05;tau=max(norm(mu,inf),norm(lam,inf));if(sigma*(tau+deta)<1)sigma=sigma;elsesigma=1.0/(tau+2*deta);endim=0;while(im<=20)if(phi1(x+rho^im*dk,sigma)-phi1(x,sigma)<eta*rho^im*dphi1(x,sigma,dk)) mk=im;break;endim=im+1;if(im==20)mk=10;endendalpha=rho^mk;x1=x+alpha*dk;[hk,gk]=cons(x1);dfk=df1(x1);[Ae,Ai]=dcons(x1);Ak=[Ae;Ai];lamu=pinv(Ak)'*dfk;if(l>0 & m>0)mu=lamu(1:l);lam=lamu(l+1:l+m);endif(l==0)mu=[];lam=lamu;endif(m==0)mu=lamu;lam=[];endsk=alpha*dk;yk=dlax(x1,mu,lam)-dlax(x,mu,lam);if(sk'*yk>0.2*sk'*Bk*sk)theta=1;elsetheta=0.8*sk'*Bk*sk/(sk'*Bk*sk-sk'*yk);endzk=theta*yk+(1-theta)*Bk*sk;Bk=Bk+zk*zk'/(sk'*zk)-(Bk*sk)*(Bk*sk)'/(sk'*Bk*sk);x=x1;k=k+1;endval=f1(x);endfunction [ d,mu,lam,val,k ] = qpsubp( dfk,Bk,Ae,hk,Ai,gk,epsilon ) %QPSUBP Summary of this function goes here% Detailed explanation goes heren=length(dfk);l=length(hk);m=length(gk);gamma=0.05;rho=0.5;sigma=0.2;ep0=0.05;mu0=0.05*zeros(l,1);lam0=0.05*zeros(m,1);d0=ones(n,1);u0=[ep0;zeros(n+l+m,1)];z0=[ep0;d0;mu0;lam0,];k=0;z=z0;ep=ep0;d=d0;mu=mu0;lam=lam0;while(1)dh=dah(ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk);if(norm(dh)<epsilon)break;endA=JacobiH(ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk);b=beta(ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk,gamma)*u0-dh;dz=pinv(A)*b;if(l>0 & m>0)de=dz(1);dd=dz(2:n+1);du=dz(n+2:n+l+1);dl=dz(n+l+2:n+l+m+1);endif(l==0)de=dz(1);dd=dz(2:n+1);dl=dz(n+2:n+m+1);endif(m==0)de=dz(1);dd=dz(2:n+1);du=dz(n+2:n+l+1);endi=0;while(i<=20)if(l>0 & m>0)dh1=dah(ep+rho^i*de,d+rho^i*dd,mu+rho^i*du,lam+rho^i*dl,dfk,Bk,Ae,hk,Ai,gk);endif(l==0)dh1=dah(ep+rho^i*de,d+rho^i*dd,mu,lam+rho^i*dl,dfk,Bk,Ae,hk,Ai,gk);endif(m==0)dh1=dah(ep+rho^i*de,d+rho^i*dd,mu+rho^i*du,lam,dfk,Bk,Ae,hk,Ai,gk);endif(norm(dh1)<=(1-sigma*(1-gamma*ep0)*rho^i)*norm(dh))mk=i;break;endi=i+1;if(i==20)mk=10;endendalpha=rho^mk;if(l>0 & m>0)ep=ep+alpha*de;d=d+alpha*dd;mu=mu+alpha*du;lam=lam+alpha*dl;endif(l==0)ep=ep+alpha*de;d=d+alpha*dd;lam=lam+alpha*dl;endif(m==0)ep=ep+alpha*de;d=d+alpha*dd;mu=mu+alpha*du;endk=k+1;endendfunction dh = dah( ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk )%DAH Summary of this function goes here% Detailed explanation goes heren=length(dfk);l=length(hk);m=length(gk);dh=zeros(n+l+m+1,1);dh(1)=ep;if(l>0 & m>0)dh(2:n+1)=Bk*d-Ae'*mu-Ai'*lam+dfk;dh(n+2:n+l+1)=hk+Ae*d;for(i=1:m)dh(n+l+1+i)=phi(ep,lam(i),gk(i)+Ai(i,:)*d);endendif(l==0)dh(2:n+1)=Bk*d-Ai'*lam+dfk;for(i=1:m)dh(n+1+i)=phi(ep,lam(i),gk(i)+Ai(i,:)*d);endendif(m==0)dh(2:n+1)=Bk*d-Ae'*mu+dfk;dh(n+2:n+l+1)=hk+Ae*d;enddh=dh(:);endfunction bet = beta( ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk,gamma )%BETA Summary of this function goes here% Detailed explanation goes heredh=dah(ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk);bet=gamma*norm(dh)*min(1,norm(dh));endfunction [ h,g ] = cons( x )%CONS Summary of this function goes here% Detailed explanation goes hereh=[ ];g=[-x(1)^2+6*x(1)-4*x(2)+11,x(1)*x(2)-3*x(2)-exp(x(1)-1)+1,x(1),x(2)]'; endfunction [ dh,dg ] = dcons( x )%DCONS Summary of this function goes here% Detailed explanation goes heredh=[ ];dg=[-2*x(1)+6,-4;x(2)-exp(x(1)-1),x(1)-1;1,0;0,1];endfunction [ dd1,dd2,v1 ] = ddv( ep,d,lam,Ai,gk)%DDV Summary of this function goes here% Detailed explanation goes herem=length(gk);dd1=zeros(m,m);dd2=zeros(m,m);v1=zeros(m,1);for(i=1:m)fm=sqrt(lam(i)^2+(gk(i)+Ai(i,:)*d)^2+2*ep^2);dd1(i,i)=1-lam(i)/fm;dd2(i,i)=1-(gk(i)+Ai(i,:)*d)/fm;v1(i)=-2*ep/fm;endendfunction df = df1( x )%DF1 Summary of this function goes here% Detailed explanation goes heredf=[2*x(1)-16,2*x(2)-10]';endfunction dl = dlax( x,mu,lam )%DLAX Summary of this function goes here% Detailed explanation goes heredf=df1(x);[Ae,Ai]=dcons(x);[m1,m2]=size(Ai);[l1,l2]=size(Ae);if(l1==0)dl=df-Ai'*lam;endif(m1==0)dl=df-Ae'*mu;endif(l1>0 & m1>0)dl=df-Ae'*mu-Ai'*lam;endendfunction f = f1( x )%F1 Summary of this function goes here% Detailed explanation goes heref=x(1)^2+x(2)^2-16*x(1)-10*x(2);endfunction p = phi( ep,a,b )%PHI Summary of this function goes here% Detailed explanation goes herep=a+b-sqrt(a^2+b^2+2*ep^2);endfunction p = phi1( x,sigma )%PHI1 Summary of this function goes here% Detailed explanation goes heref=f1(x);[h,g]=cons(x);gn=max(-g,0);l0=length(h);m0=length(g);if(l0==0)p=f+1.0/sigma*norm(gn,1);endif(m0==0)p=f+1.0/sigma*norm(h,1);endif(l0>0 & m0>0)p=f+1.0/sigma*(norm(h,1)+norm(gn,1)); endendfunction dp = dphi1( x,sigma,d )%DPHI1 Summary of this function goes here% Detailed explanation goes heredf=df1(x);[h,g]=cons(x);gn=max(-g,0);l0=length(h);m0=length(g);if(l0==0)dp=df'*d-1.0/sigma*norm(gn,1);endif(m0==0)dp=df'*d-1.0/sigma*norm(h,1);endif(l0>0 & m0>0)dp=df'*d-1.0/sigma*(norm(h,1)+norm(gn,1)); endendfunction A = JacobiH( ep,d,mu,lam,dfk,Bk,Ae,hk,Ai,gk )%JACOBIH Summary of this function goes here% Detailed explanation goes heren=length(dfk);l=length(hk);m=length(gk);A=zeros(n+l+m+1,n+l+m+1);[dd1,dd2,v1]=ddv(ep,d,lam,Ai,gk);if(l>0 & m>0)A=[1, zeros(1,n), zeros(1,l), zeros(1,m);zeros(n,1), Bk, -Ae', -Ai';zeros(l,1), Ae, zeros(l,l), zeros(l,m);v1, dd2*Ai, zeros(m,l), dd1]; endif(l==0)A=[1, zeros(1,n), zeros(1,m);zeros(n,1), Bk, -Ai';v1, dd2*Ai, dd1];endif(m==0)A=[1, zeros(1,n), zeros(1,l);zeros(n,1), Bk, -Ae';zeros(l,1), Ae, zeros(l,l)];endend（2）代码运行结果如下：解：（1）MATLAB代码如下：clc;close all;clear all;f=[1 1 1 1 1 1 1 1];A=[1 0 0 0.5;0 1 0.2 0.3;0 0.1 1 0.2];Aeq=[A,-A];beq=[-1 0.2 1]';lb=zeros(8,1);x=linprog(f,[],[],Aeq,beq,lb);x=[x(1)-x(5) x(2)-x(6) x(3)-x(7) x(4)-x(8)]' （2）代码运行结果如下：。
实用最优化方法大连理工课后答案
实用最优化方法大连理工课后答案
1．下列情况引起的误差是系统误差还是偶然误差？
(1)砝码锈蚀（系统误差）
(2)称重时试样吸收了空气中的水分；（系统误差）
(3)滴定管读数时末位数字估计不准嘶；（偶然误差）
(4)滴定剂中台有少量待测组分；睬统误差）
(5)标定用的基准物Na2C03在保存过程中吸收了水分：（系统误整）
(6)称量过程中天平零点由于环境条件的变化稍有变动（偶然误差）
2．什么是误差？什么是偏差？有什么区别和联系？
误差是测量值与真值之差偏差是单次测量值与n次测量平均值之差,误差是用测量位与真实值作比较，衡量准确度的高低，偏差是用测定值与平均位作比较，用于衡量青密度的大小，准确度高则精密度一定高，精密度高准确度不一定高。
大连理工大学概率上机作业
大连理工大学概率上机作业————————————————————————————————作者: ————————————————————————————————日期：ﻩ第一次上机作业1.利用Matlａb自带命令产生100０个均匀随机变量服从Ｕ(０，1）。
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０．1455 0．４909 0.０４30 0.4468０.1419 0.8２3５0.49８4 ０．35０0 0.1２9９0.８2５８0.１36１0.4８93 0.1690 0.３０63０.4２18 0.6948０.9597 0.１966０.５６8８０.5383 0.8693 ０.3３７7 0.6491 0.508５0.9157 0.3171０.３4０4 0.2５１1 0．46９4 0.996１0.579７0.9０0１0．7３17 0.51０80.7922 0．９50２0.5853 0.61600.０1190.07８20．54９９０.３692 0.6４77 ０．81760.9５950.0344 ０.2２38 ０.4733 0.３3７1 0.4427０.1450 ０．1１120.4509 0.794８Colｕmns １1 throｕｇh 200.6443 0.31１10．08５5 ０.0377 0.03050.0596 ０．173４0.９51６0.０3２6 0．25180.３7８6 ０.9２3４0.262５0．８8５2 ０.74４1０.68200．39０9 0.920３0.５6120.29040.８116 0.4302 0.8010 ０.9１3３0．500０0．04２4 0.831４0.0527０.８819 0.6171０．５328 0．18480．0292 ０.79620.４7990.07１４0.8034 ０.７379 0.6６9２0.265３0.3５07 0.9049 ０.92８9 0.0987 0.９047０．５21６0.06050.26９10.１９0４0.824４0.9390 0．9７97 0．７303 0.26１９０.609９0.０９6７0．39９3０.４228０．3689 0．982７0.８759 ０.4389 0.4886 ０.３354 0.6177 0.８18１0．5269 0.54790.4607 ０.730２0.55020.11１1 ０.5785 0．67９7 ０.8５94 0．81750．4１680.94２７0.9816 0.３4390.６２２５0.25８1 0.23７30.1366 0.8055 0.７２24 0.６5690.4177 0.1５6４0.58４1０．５870 0.4087 0.458８０．7212 0.5７6７0.1499０．6２80 ０.９83１０．8555 0.1０780．20７70.5949 ０.963１0.106８0.18２９０．６5９6 0.2920 0.3０１5 0.6448 0.９06３0．3012 ０．2622 0．5４680.6５38 ０.２３99０.５186 0.4３1７0.７0１1 0.３763 0.8797０.4709 0.6０2８0．52110.494２0．88６5 0.９７3００.0155 0.６663 0.190９0．817８0.230５0．7112 0.231６0.７79１０.02８70.6490 ０.9841 0．5３91 0．4283 ０.260７0．844３０.2217 ０.488９０.７１50 0．4899 ０.8003 0.1６72 0.6９８１0.４８20 0.59４4０．19４8 0.1174 ０.６241 0．90370．１6790．4５38 0.106２0.666５0．1206 0.02２５0.2２5９0.2９67 0.67９1 0.89０9 0．97８7 0.４３2４0.3724 0.1781 0.５8９５0．425３０.1707 ０．３188 ０.３９55 0.3342 ０.7127 ０.82５3 ０.1981 ０.１280 0．2262 0.31２7０.2277 0．4242 0.3674 0.６987 0.5００5 0.０835 ０.48970.99９1 ０.３846 0.１6150.43５7 0.5０７9 ０.988００.197８0.471１0.1332 ０.3395０.171１0．5830 0.１７88Coluｍnｓ２1thrｏugh 300.4２２９０.７7８8 0.２54８0．17５9 0．６476 0.5８22 ０．4046 0.3477 0．８2１７0.5１４4０．0942 0．42３５0.22４0 ０．7２18 ０．67900.5４070.44８4 ０．1500 ０.42９９0.884３0.598５0．09080.6６７80．473５0.63５8 0．86990．3658 0．5861 ０.88780．58８00.47090.2６65 0.8４４4 0.１527 0.945２0.26480.７63５0.2621 ０．3912 0.1548０.69５9 0.1５3７０.３44５0．34１1０.２089 0.３１８1 ０.６27９0.044５0．7691 0.１999０.６999０．28１0 ０.780５０.６07４0.70930.119２0．77２0 ０.7５49 0．39６8 ０．40７0０.638５0.440１０.6753 0.19170.２３6２0.9398 0.932９0．2428 ０．8085 ０.74870.033６0.5２7１0.０0６7 0.73840．11940.64560.9727 0.442４０．7551 0.8２5６０.０６88 0.４５7４０.6022 0.2４２80.６073 0.4795 0.192００.687８０.3７740.79000.31９6 0.875４０．386８0.9１74 ０.４５０1 0．639３0.13８9０．359２0.2160 0.3185０.53090.5181 ０.9１600.２6９1 ０.4５87０.54４7 0.69６３０.7３63 0.7904 0.53410.6５４4 0.9436 ０.00１2 0．76５5 ０.6619 0.64730.09３8 0.3947 0.９４9３0.090０0.４0７6 0.6377 0.４6２4 0.1887 0．７70３０．5４39 0.52５4 0．6834 0.３2７60．１11７0.8200 0.9５770.4２4３0.287５0.3502 0.７2１0 0.５3030．7040 0．6713 0.13６３0.７18４0.24０７0.46090.0911 0.66２0 0.5225 ０.8611 0.4423 0.4３8６0.67８70．9６8６0.6７61 0.7７0２0.５762 0．41６20.９937 0.４849 0.０196 ０．8３35 0.49520.53１3 0.２89１0.3２25 0．６83４０.８４１９0.２18７０．39３50.33０9 ０.７689 ０．１8970.3251 0.671８0.7847 ０.５466 ０.８32９０．１０58 ０.67１４0．4243 ０.１67３0.495００.1０5６0．6９51０.４7１4 0.4257 ０.２56４０．109７0．７４１3 ０．２7０3 0.８620 0.14７60．61１0 ０．0６８０0.0358０．64４4 ０.６13５0．063６0．520１0．1971 0.9899 0．0５５0Cｏlumｎs 31 thｒｏugh 400．8５０７0．7３8６0.55２３0．１２３９０．737８0.55９0 ０.1781 0.894９0．6３11 ０.6９25０．５60６0.5８６00.62９９０.4904 0．0６3４０.8541 ０.３5９6 0.0715０.0８9９0.55６70.９２96 0.246７0.０320０.8530 0.860４0.3４79 0.０567 0.２425 ０.080９０.3９65０.696７0.６664 ０．6１4７0.873９0.93４４0.4460 0.5２19 ０.05３8 0.777２0.0６1６０．５８２80.０8３50.３624 0.2703 0.9844 0.0542 ０.３358 0．441７0．９０51 0．78020.８1５4 0．62６00.０４９50．2085 0.８589 0.1771０．１７57０.0１330．53380.3３760.87９0 0.６６09 0．４8９6 0.５650 0．7856 0.6６28 0．２08９０.８９7２0．109２0.６0790.988９0.7298 0.19２50．6403 0．51340.３３0８０．90520．19６7 0.8２5８0.7４13０.０0050.89０80.12310.４17０0.17760.8985 ０.67５4 0.09３４0.3３81 0.１048０．8６5４０.98230.２05５0.2060 0.39860.1182 0.4６８5 0．3０74 0．２９40 0.1２79０.６126 0.76９０0．１４6５0.9479０.１３3９0．9884 0.9１2１0.4５61 0.74６3 0.54９50.99０0０.５81４0.1891 0.0８21 ０．030９０.54000.1０４００.１017 0.0103 0.4８520.5277 0.９２８3 ０.0４２7 0．１０57０．9３９1 0.７０69 ０．74５５0.9954 0.０484 0.89０50.４795 0.5801 0．6352 0.142０0.３０1３0.9９95 ０．7363 0.3321 0.６6７９0.7990０.8013 0.０170 0.2819 0．１665 0.295５0.287８0.5619 0.２973 0．6035 0.7343０.2278 0.1209 ０.５386 0.62１00.3３２9 ０.41４5 0.18４20.0620０.526１０.05１30.49８1 0．86２7 0.6952 ０．5７3７0.4671 0．4648 ０.5972 0.２982 ０.7２97０.07２90.9009０.4843 0.４９91 0.0521 0.648２0.7640 ０.2９99 0.0464 0.7073０．088５０.57４70.８44９0.53５８0．9312 0.02５2 ０.81820.134１0.505４０.78１4 0.7984０.８452 0.20940．44５2 0.7287 0.8４22 0.１０0２0.２12６0．76１4０.288０0.9４30Coｌuｍns 41 ｔhｒough 5０0．６8３7 ０.789４0．1123 ０.６733 0．09８６0.9879 0．5９７5 0．７5９3０.809２0.7５190.1３２1 0.367７0.７8４４0.4２9６0.142００.1704 0.33５3 ０.7４０6 0.7486 0.２2870.7２27 ０．2060 0.2916 0.4517 0．１６83 0.2578 ０.２9９2 0．74370.120２０．０6420．110４0.0８67 0.6０350.6０99 0.1９620.3968 ０.4526 ０.１0590.５２50 0．7６730.11７５０.77１９0.9644 0．0594 0.３17５0.0740 0.42２6 ０.68１６０.３２５8 0.6712０.6407 0.2057 0.４3250．3１５8 0.３1640．６841 0.3５9６0.46330.５４64 0．７15２0.3２88 ０.３8８３0.6948 0.7727 0.2176 0.4024 0.5583 0.21２２0.39８9 0.64210.65380.55１8 0.7５810.６９64 0.2５１00.９8２8 0.74250.0985０.4１51 0.41900.74９1 0.2290 0.43２6 0.１２5３0．8929 0.4022 0.42４3 ０．8２360.18０7 0．39080.５８320.６４１9 0.６5550．１３02 0.703２0.６２０7 0．4294 0.17５0 0.25５4 0.81６1０.7400０.48４５０.１0980．0９24 0.５557 0.1544 0．1２49 0.1636 0.02０5 0.317４0.23４8 0.15180．933８0.０078０．1844 0．3813 0.0244 0．66６００.9237 0.８１45０.7350 0.78１9０．18７5 0．４2３１0.２１200．1611 0.２902 ０.８944 0．６５3７0.７89１0.97060.10060．2662 0.655６0.0７7３０.7581０．31７5 ０.5166 0.9３２６0.８5２3０.８669 ０.29410．７978 0．7229 ０.91３８0.８７11 0.６5370.70２7０．1６３5 0.５056０.08620.2３7４0.48760.5312０.7０6７0.３5080.9569 0.1536 0.９２11 0.63570．３664 0.５309 0.769０0.10880.55７8 ０.68550.9３57 0．95３50.７94７0.９509０．３6９2 0.0915 0.39６0 0.6318０.３13４０.2９４1 0.４５79 ０.540９0.577４0.４44０0.68５0 0．405３0.2729０.126５0.1662 ０.530６０.240５0.6797０.4400 ０.06000.5９７9 ０.１04８0.0３72 0.13４3 0．６225 0.832４０.76390.03660．257６0.86672.参考课本综合例题2.5.４和2．5.5中的方法，模拟产生1０00个随机变量，使其服从参数为2的指数分布,进而计算这1000个随机数的均值和方差。
优化方法MATLAB编程——大连理工大学
The optimal solution is 0.000000.
The optimal "x" is "ans".
ans =
1.0000 1.0000 1.0000 1.0000 可以看出，用Newton法经过6次迭代就能求出最优解。
3. BFGS法 源程序如下： function zy_x=di2tiBFGS(x) %该函数用来解大作业第二题。 x0=x; yimuxulong=1e-5; k=0; g0=g(x0); H0=eye(4);s0=-H0*g0;
3
s0=-g0;
大连理工大学优化方法上机大作业
end end x=x0+lanmed*s0; x0=x; g0=g(x); s0=-g0; k=k+1; end end zy_x=x; zyj=f(x); fprintf('after %d iterations,obtain the optimal solution.\n\nThe optimal solution is %f.\n\n The optimal "x" is "ans".\n',k,zyj);
4
大连理工大学优化方法上机大作业
>> x=[-1.2 1 -1.2 1]'; >> di2titidu(x) after 5945 iterations,obtain the optimal solution.
The optimal solution is 0.000000.
The optimal "x" is "ans".
大连理工大学优化方法上机大作业
大连理工大学优化方法上机大作业
学院：专业：班级：学号：姓名：上机大作业1：1.最速下降法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = steepest(x0,eps) gk = grad(x0);res = norm(gk);k = 0;while res > eps && k<=1000dk = -gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + *ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;gk = grad(xk);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=steepest(x0,eps)2.牛顿法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad2(x)g = zeros(2,2);g(1,1)=2+400*(3*x(1)^2-x(2));g(1,2)=-400*x(1);g(2,1)=-400*x(1);g(2,2)=200;endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = newton(x0,eps)gk = grad(x0);bk = [grad2(x0)]^(-1);res = norm(gk);k = 0;while res > eps && k<=1000dk=-bk*gk;xk=x0+dk;k = k+1;x0 = xk;gk = grad(xk);bk = [grad2(xk)]^(-1);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x1=newton(x0,eps)--The 1-th iter, the residual is--The 2-th iter, the residual isx1 =法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = bfgs(x0,eps) g0 = grad(x0);gk=g0;res = norm(gk);Hk=eye(2);k = 0;while res > eps && k<=1000dk = -Hk*gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + *ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;fa0=xk-x0;x0 = xk;go=gk;gk = grad(xk);y0=gk-g0;Hk=((eye(2)-fa0*(y0)')/((fa0)'*(y0)))*((eye(2)-(y0)*(fa0)')/((fa0)'*(y0)))+(fa0*(fa 0)')/((fa0)'*(y0));res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res);endx_star = xk;End>> clear>> x0=[0,0]';>> eps=1e-4;>> x=bfgs(x0,eps)4.共轭梯度法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star =CG(x0,eps) gk = grad(x0);res = norm(gk);k = 0;dk = -gk;while res > eps && k<=1000 ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + *ak*slope ak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;g0=gk;gk = grad(xk);res = norm(gk);p=(gk/g0)^2;dk1=dk;dk=-gk+p*dk1;fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=CG(x0,eps)上机大作业2：function f= obj(x)f=4*x(1)-x(2)^2-12;endfunction [h,g] =constrains(x) h=x(1)^2+x(2)^2-25;g=zeros(3,1);g(1)=-10*x(1)+x(1)^2-10*x(2)+x(2)^2+34;g(2)=-x(1);g(3)=-x(2);endfunction f=alobj(x) %拉格朗日增广函数%N_equ等式约束个数?%N_inequ不等式约束个数N_equ=1;N_inequ=3;global r_al pena;%全局变量h_equ=0;h_inequ=0;[h,g]=constrains(x);%等式约束部分?for i=1:N_equh_equ=h_equ+h(i)*r_al(i)+(pena/2)*h(i).^2;end%不等式约束部分for i=1:N_inequh_inequ=h_inequ+pena)*(max(0,(r_al(i)+pena*g(i))).^2-r_al(i).^2); end%拉格朗日增广函数值f=obj(x)+h_equ+h_inequ;function f=compare(x)global r_al pena N_equ N_inequ;N_equ=1;N_inequ=3;h_inequ=zeros(3,1);[h,g]=constrains(x);%等式部分for i=1:1h_equ=abs(h(i));end%不等式部分for i=1:3h_inequ=abs(max(g(i),-r_al(i+1)/pena));endh1 = max(h_inequ);f= max(abs(h_equ),h1); %sqrt(h_equ+h_inequ);function [ x,fmin,k] =almain(x_al)%本程序为拉格朗日乘子算法示例算法%函数输入：% x_al:初始迭代点% r_al：初始拉格朗日乘子N-equ:等式约束个数N_inequ:不等式约束个数?%函数输出% X：最优函数点FVAL:最优函数值%============================程序开始================================ global r_al pena ; %参数（全局变量）pena=10; %惩罚系数r_al=[1,1,1,1];c_scale=2; %乘法系数乘数cta=; %下降标准系数e_al=1e-4; %误差控制范围max_itera=25;out_itera=1; %迭代次数%===========================算法迭代开始============================= while out_itera<max_iterax_al0=x_al;r_al0=r_al;%判断函数?compareFlag=compare(x_al0);%无约束的拟牛顿法BFGS[X,fmin]=fminunc(@alobj,x_al0);x_al=X; %得到新迭代点%判断停止条件?if compare(x_al)<e_aldisp('we get the opt point');breakend%c判断函数下降度?if compare(x_al)<cta*compareFlagpena=1*pena; %可以根据需要修改惩罚系数变量elsepena=min(1000,c_scale*pena); %%乘法系数最大1000disp('pena=2*pena');end%%?更新拉格朗日乘子[h,g]=constrains(x_al);for i=1:1%%等式约束部分r_al(i)= r_al0(i)+pena*h(i);endfor i=1:3%%不等式约束部分r_al(i+1)=max(0,(r_al0(i+1)+pena*g(i)));endout_itera=out_itera+1;end%+++++++++++++++++++++++++++迭代结束+++++++++++++++++++++++++++++++++ disp('the iteration number');k=out_itera;disp('the value of constrains'); compare(x_al)disp('the opt point');x=x_al;fmin=obj(X);>> clear>> x_al=[0,0];>> [x,fmin,k]=almain(x_al)上机大作业3： 1、>> clear alln=3; c=[-3,-1,-3]'; A=[2,1,1;1,2,3;2,2,1;-1,0,0;0,-1,0;0,0,-1];b=[2,5,6,0,0,0]';cvx_beginvariable x(n)minimize( c'*x)subject toA*x<=bcvx_endCalling SDPT3 : 6 variables, 3 equality constraints------------------------------------------------------------num. of constraints = 3dim. of linear var = 6*******************************************************************SDPT3: Infeasible path-following algorithms*******************************************************************version predcorr gam expon scale_dataNT 1 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime -------------------------------------------------------------------0|||+01|+00|+02|+01 +00| 0:0:00| chol 1 11|||||+01|+00 +01| 0:0:01| chol 1 12|||||+00|+00 +01| 0:0:01| chol 1 13|||||+00|+00 +00| 0:0:01| chol 1 14||||||+00 +00| 0:0:01| chol 1 15||||||+00 +00| 0:0:01| chol 1 16||||||+00 +00| 0:0:01| chol 1 17||||||+00 +00| 0:0:01| chol 1 18||||||+00 +00| 0:0:01|stop: max(relative gap, infeasibilities) <------------------------------------------------------------------- number of iterations = 8primal objective value = +00dual objective value = +00gap := trace(XZ) =relative gap =actual relative gap =rel. primal infeas (scaled problem) =rel. dual " " " =rel. primal infeas (unscaled problem) = +00rel. dual " " " = +00norm(X), norm(y), norm(Z) = +00, +00, +00norm(A), norm(b), norm(C) = +00, +00, +00Total CPU time (secs) =CPU time per iteration =termination code = 0DIMACS: +00 +00-------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval):2、>> clear alln=2; c=[-2,-4]'; G=[,0;0,1]; A=[1,1;-1,0;0,-1]; b=[1,0,0]'; cvx_beginvariable x(n)minimize( x'*G*x+c'*x)subject toA*x<=bcvx_endCalling SDPT3 : 7 variables, 3 equality constraintsFor improved efficiency, SDPT3 is solving the dual problem.------------------------------------------------------------num. of constraints = 3dim. of socp var = 4, num. of socp blk = 1dim. of linear var = 3*******************************************************************SDPT3: Infeasible path-following algorithms*******************************************************************version predcorr gam expon scale_dataNT 1 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime -------------------------------------------------------------------0||||+00|+02| +01 +00| 0:0:00| chol 1 11|||||+01| +00 | 0:0:00| chol 1 12|||||+00| +00 | 0:0:00| chol 1 13|||||| | 0:0:00| chol 1 14|||||| | 0:0:00| chol 1 15|||||| | 0:0:00| chol 1 16|||||| | 0:0:00| chol 1 17|||||| | 0:0:00| chol 1 18|||||| | 0:0:00| chol 1 19|||||| | 0:0:00| chol 1 110|||||| | 0:0:00| chol 2 211|||||| | 0:0:00| chol 2 212|||||| | 0:0:00| chol 2 213|||||| | 0:0:00| chol 2 214|||||| | 0:0:00|stop: max(relative gap, infeasibilities) <------------------------------------------------------------------- number of iterations = 14primal objective value =dual objective value =gap := trace(XZ) =relative gap =actual relative gap =rel. primal infeas (scaled problem) =rel. dual " " " =rel. primal infeas (unscaled problem) = +00rel. dual " " " = +00norm(X), norm(y), norm(Z) = +00, +00, +00norm(A), norm(b), norm(C) = +00, +00, +00Total CPU time (secs) =CPU time per iteration =termination code = 0DIMACS: +00 +00-------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval): -3。
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2016年大连理工大学优化方法上机大作业-标准化文件发布号：（9456-EUATWK-MWUB-WUNN-INNUL-DDQTY-KII2016年大连理工大学优化方法上机大作业学院：专业：班级：学号：姓名：上机大作业1：1.最速下降法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = steepest(x0,eps)gk = grad(x0);res = norm(gk);k = 0;while res > eps && k<=1000dk = -gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;gk = grad(xk);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=steepest(x0,eps)2.牛顿法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad2(x)g = zeros(2,2);g(1,1)=2+400*(3*x(1)^2-x(2));g(1,2)=-400*x(1);g(2,1)=-400*x(1);g(2,2)=200;endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2));g(2) = 200*(x(2)-x(1)^2);endfunction x_star = newton(x0,eps)gk = grad(x0);bk = [grad2(x0)]^(-1);res = norm(gk);k = 0;while res > eps && k<=1000dk=-bk*gk;xk=x0+dk;k = k+1;x0 = xk;gk = grad(xk);bk = [grad2(xk)]^(-1);res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x1=newton(x0,eps)--The 1-th iter, the residual is 447.213595--The 2-th iter, the residual is 0.000000x1 =1.00001.00003.BFGS法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star = bfgs(x0,eps)g0 = grad(x0);gk=g0;res = norm(gk);Hk=eye(2);k = 0;while res > eps && k<=1000dk = -Hk*gk;ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;fa0=xk-x0;x0 = xk;go=gk;gk = grad(xk);y0=gk-g0;Hk=((eye(2)-fa0*(y0)')/((fa0)'*(y0)))*((eye(2)-(y0)*(fa0)')/((fa0)'*(y0)))+(fa0*(fa0)')/((fa0)'*(y0)); res = norm(gk);fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;End>> clear>> x0=[0,0]';>> eps=1e-4;>> x=bfgs(x0,eps)4.共轭梯度法：function f = fun(x)f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2; endfunction g = grad(x)g = zeros(2,1);g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);endfunction x_star =CG(x0,eps)gk = grad(x0);res = norm(gk);k = 0;dk = -gk;while res > eps && k<=1000ak =1; f0 = fun(x0);f1 = fun(x0+ak*dk);slope = dot(gk,dk);while f1 > f0 + 0.1*ak*slopeak = ak/4;xk = x0 + ak*dk;f1 = fun(xk);endk = k+1;x0 = xk;g0=gk;gk = grad(xk);res = norm(gk);p=(gk/g0)^2;dk1=dk;dk=-gk+p*dk1;fprintf('--The %d-th iter, the residual is %f\n',k,res); endx_star = xk;end>> clear>> x0=[0,0]';>> eps=1e-4;>> x=CG(x0,eps)上机大作业2：function f= obj(x)f=4*x(1)-x(2)^2-12;endfunction [h,g] =constrains(x)h=x(1)^2+x(2)^2-25;g=zeros(3,1);g(1)=-10*x(1)+x(1)^2-10*x(2)+x(2)^2+34; g(2)=-x(1);g(3)=-x(2);endfunction f=alobj(x) %拉格朗日增广函数%N_equ等式约束个数?%N_inequ不等式约束个数N_equ=1;N_inequ=3;global r_al pena;%全局变量h_equ=0;h_inequ=0;[h,g]=constrains(x);%等式约束部分?for i=1:N_equh_equ=h_equ+h(i)*r_al(i)+(pena/2)*h(i).^2;end%不等式约束部分for i=1:N_inequh_inequ=h_inequ+(0.5/pena)*(max(0,(r_al(i)+pena*g(i))).^2-r_al(i).^2);end%拉格朗日增广函数值f=obj(x)+h_equ+h_inequ;function f=compare(x)global r_al pena N_equ N_inequ;N_equ=1;N_inequ=3;h_inequ=zeros(3,1);[h,g]=constrains(x);%等式部分for i=1:1h_equ=abs(h(i));end%不等式部分for i=1:3h_inequ=abs(max(g(i),-r_al(i+1)/pena));endh1 = max(h_inequ);f= max(abs(h_equ),h1); %sqrt(h_equ+h_inequ);function [ x,fmin,k] =almain(x_al)%本程序为拉格朗日乘子算法示例算法%函数输入：% x_al:初始迭代点% r_al：初始拉格朗日乘子N-equ:等式约束个数N_inequ:不等式约束个数%函数输出% X：最优函数点FVAL:最优函数值%============================程序开始================================ global r_al pena ; %参数（全局变量）pena=10; %惩罚系数r_al=[1,1,1,1];c_scale=2; %乘法系数乘数cta=0.5; %下降标准系数e_al=1e-4; %误差控制范围max_itera=25;out_itera=1; %迭代次数%===========================算法迭代开始============================= while out_itera<max_iterax_al0=x_al;r_al0=r_al;%判断函数?compareFlag=compare(x_al0);%无约束的拟牛顿法BFGS[X,fmin]=fminunc(@alobj,x_al0);x_al=X; %得到新迭代点%判断停止条件?if compare(x_al)<e_aldisp('we get the opt point');breakend%c判断函数下降度?if compare(x_al)<cta*compareFlagpena=1*pena; %可以根据需要修改惩罚系数变量elsepena=min(1000,c_scale*pena); %%乘法系数最大1000disp('pena=2*pena');end%%更新拉格朗日乘子[h,g]=constrains(x_al);for i=1:1%%等式约束部分r_al(i)= r_al0(i)+pena*h(i);endfor i=1:3%%不等式约束部分r_al(i+1)=max(0,(r_al0(i+1)+pena*g(i)));endout_itera=out_itera+1;end%+++++++++++++++++++++++++++迭代结束+++++++++++++++++++++++++++++++++ disp('the iteration number');k=out_itera;disp('the value of constrains');compare(x_al)disp('the opt point');x=x_al;fmin=obj(X);>> clear>> x_al=[0,0];>> [x,fmin,k]=almain(x_al)上机大作业3：1、>> clear alln=3; c=[-3,-1,-3]'; A=[2,1,1;1,2,3;2,2,1;-1,0,0;0,-1,0;0,0,-1];b=[2,5,6,0,0,0]';cvx_beginvariable x(n)minimize( c'*x)subject toA*x<=bcvx_endCalling SDPT3 4.0: 6 variables, 3 equality constraints------------------------------------------------------------num. of constraints = 3dim. of linear var = 6******************************************************************* SDPT3: Infeasible path-following algorithms******************************************************************* version predcorr gam expon scale_dataNT 1 0.000 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime-------------------------------------------------------------------0|0.000|0.000|1.1e+01|5.1e+00|6.0e+02|-7.000000e+01 0.000000e+00| 0:0:00| chol 1 11|0.912|1.000|9.4e-01|4.6e-02|6.5e+01|-5.606627e+00 -2.967567e+01| 0:0:01| chol 1 12|1.000|1.000|1.3e-07|4.6e-03|8.5e+00|-2.723981e+00 -1.113509e+01| 0:0:01| chol 1 13|1.000|0.961|2.3e-08|6.2e-04|1.8e+00|-4.348354e+00 -6.122853e+00| 0:0:01| chol 1 14|0.881|1.000|2.2e-08|4.6e-05|3.7e-01|-5.255152e+00 -5.622375e+00| 0:0:01| chol 1 15|0.995|0.962|1.6e-09|6.2e-06|1.5e-02|-5.394782e+00 -5.409213e+00| 0:0:01| chol 1 16|0.989|0.989|2.7e-10|5.2e-07|1.7e-04|-5.399940e+00 -5.400100e+00| 0:0:01| chol 1 17|0.989|0.989|5.3e-11|5.8e-09|1.8e-06|-5.399999e+00 -5.400001e+00| 0:0:01| chol 1 18|1.000|0.994|2.8e-13|4.3e-11|2.7e-08|-5.400000e+00 -5.400000e+00| 0:0:01|stop: max(relative gap, infeasibilities) < 1.49e-08-------------------------------------------------------------------number of iterations = 8primal objective value = -5.39999999e+00dual objective value = -5.40000002e+00gap := trace(XZ) = 2.66e-08relative gap = 2.26e-09actual relative gap = 2.21e-09rel. primal infeas (scaled problem) = 2.77e-13rel. dual " " " = 4.31e-11rel. primal infeas (unscaled problem) = 0.00e+00rel. dual " " " = 0.00e+00norm(X), norm(y), norm(Z) = 4.3e+00, 1.3e+00, 1.9e+00norm(A), norm(b), norm(C) = 6.7e+00, 9.1e+00, 5.4e+00Total CPU time (secs) = 0.71CPU time per iteration = 0.09termination code = 0DIMACS: 3.6e-13 0.0e+00 5.8e-11 0.0e+00 2.2e-09 2.3e-09 -------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval): -5.42、>> clear alln=2; c=[-2,-4]'; G=[0.5,0;0,1];A=[1,1;-1,0;0,-1]; b=[1,0,0]';cvx_beginvariable x(n)minimize( x'*G*x+c'*x)subject toA*x<=bcvx_endCalling SDPT3 4.0: 7 variables, 3 equality constraintsFor improved efficiency, SDPT3 is solving the dual problem.------------------------------------------------------------num. of constraints = 3dim. of socp var = 4, num. of socp blk = 1dim. of linear var = 3******************************************************************* SDPT3: Infeasible path-following algorithms*******************************************************************version predcorr gam expon scale_dataNT 1 0.000 1 0it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime-------------------------------------------------------------------0|0.000|0.000|8.0e-01|6.5e+00|3.1e+02| 1.000000e+01 0.000000e+00| 0:0:00| chol 1 11|1.000|0.987|4.3e-07|1.5e-01|1.6e+01| 9.043148e+00 -2.714056e-01| 0:0:00| chol 1 12|1.000|1.000|2.6e-07|7.6e-03|1.4e+00| 1.234938e+00 -5.011630e-02| 0:0:00| chol 1 13|1.000|1.000|2.4e-07|7.6e-04|3.0e-01| 4.166959e-01 1.181563e-01| 0:0:00| chol 1 1 4|0.892|0.877|6.4e-08|1.6e-04|5.2e-02| 2.773022e-01 2.265122e-01| 0:0:00| chol 1 1 5|1.000|1.000|1.0e-08|7.6e-06|1.5e-02| 2.579468e-01 2.427203e-01| 0:0:00| chol 1 1 6|0.905|0.904|3.1e-09|1.4e-06|2.3e-03| 2.511936e-01 2.488619e-01| 0:0:00| chol 1 1 7|1.000|1.000|6.1e-09|7.7e-08|6.6e-04| 2.503336e-01 2.496718e-01| 0:0:00| chol 1 1 8|0.903|0.903|1.8e-09|1.5e-08|1.0e-04| 2.500507e-01 2.499497e-01| 0:0:00| chol 1 1 9|1.000|1.000|4.9e-10|3.5e-10|2.9e-05| 2.500143e-01 2.499857e-01| 0:0:00| chol 1 1 10|0.904|0.904|4.7e-11|1.3e-10|4.4e-06| 2.500022e-01 2.499978e-01| 0:0:00| chol 2 211|1.000|1.000|2.3e-12|9.4e-12|1.2e-06| 2.500006e-01 2.499994e-01| 0:0:00| chol 2 212|1.000|1.000|4.7e-13|1.0e-12|1.8e-07| 2.500001e-01 2.499999e-01| 0:0:00| chol 2 213|1.000|1.000|2.0e-12|1.0e-12|4.2e-08| 2.500000e-01 2.500000e-01| 0:0:00| chol 2 214|1.000|1.000|2.6e-12|1.0e-12|7.3e-09| 2.500000e-01 2.500000e-01| 0:0:00|stop: max(relative gap, infeasibilities) < 1.49e-08-------------------------------------------------------------------number of iterations = 14primal objective value = 2.50000004e-01dual objective value = 2.49999996e-01gap := trace(XZ) = 7.29e-09relative gap = 4.86e-09actual relative gap = 4.86e-09rel. primal infeas (scaled problem) = 2.63e-12rel. dual " " " = 1.00e-12rel. primal infeas (unscaled problem) = 0.00e+00rel. dual " " " = 0.00e+00norm(X), norm(y), norm(Z) = 3.2e+00, 1.5e+00, 1.9e+00norm(A), norm(b), norm(C) = 3.9e+00, 4.2e+00, 2.6e+00Total CPU time (secs) = 0.36CPU time per iteration = 0.03termination code = 0DIMACS: 3.7e-12 0.0e+00 1.3e-12 0.0e+00 4.9e-09 4.9e-09 -------------------------------------------------------------------------------------------------------------------------------Status: SolvedOptimal value (cvx_optval): -3。