大连理工大学优化方法上机作业

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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。

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。

大连理工大学概率上机作业

大连理工大学概率上机作业第一次上机作业1.利用Matlab自带命令产生1000个均匀随机变量服从U(0,1)。

>> unifrnd(0,1,20,50)ans =Columns 1 through 100.8147 0.6557 0.4387 0.7513 0.3517 0.1622 0.1067 0.8530 0.7803 0.54700.9058 0.0357 0.3816 0.2551 0.8308 0.7943 0.9619 0.6221 0.3897 0.29630.1270 0.8491 0.7655 0.5060 0.5853 0.3112 0.0046 0.3510 0.2417 0.74470.9134 0.9340 0.7952 0.6991 0.5497 0.5285 0.7749 0.5132 0.4039 0.18900.6324 0.6787 0.1869 0.8909 0.9172 0.1656 0.8173 0.4018 0.0965 0.68680.0975 0.7577 0.4898 0.9593 0.2858 0.6020 0.8687 0.0760 0.1320 0.18350.2785 0.7431 0.4456 0.5472 0.7572 0.2630 0.0844 0.2399 0.9421 0.36850.5469 0.3922 0.6463 0.1386 0.7537 0.6541 0.3998 0.1233 0.9561 0.62560.9575 0.6555 0.7094 0.1493 0.3804 0.6892 0.2599 0.1839 0.5752 0.78020.9649 0.1712 0.7547 0.2575 0.5678 0.7482 0.8001 0.2400 0.0598 0.08110.1576 0.7060 0.2760 0.8407 0.0759 0.4505 0.4314 0.4173 0.2348 0.92940.9706 0.0318 0.6797 0.2543 0.0540 0.0838 0.9106 0.0497 0.3532 0.77570.9572 0.2769 0.6551 0.8143 0.5308 0.2290 0.1818 0.9027 0.8212 0.48680.4854 0.0462 0.1626 0.2435 0.7792 0.9133 0.2638 0.9448 0.0154 0.43590.8003 0.0971 0.1190 0.9293 0.9340 0.1524 0.1455 0.4909 0.0430 0.44680.1419 0.8235 0.4984 0.3500 0.1299 0.8258 0.1361 0.4893 0.1690 0.30630.4218 0.6948 0.9597 0.1966 0.5688 0.5383 0.8693 0.3377 0.6491 0.50850.9157 0.3171 0.3404 0.2511 0.4694 0.9961 0.5797 0.9001 0.7317 0.51080.7922 0.9502 0.5853 0.6160 0.0119 0.0782 0.5499 0.3692 0.6477 0.81760.9595 0.0344 0.2238 0.4733 0.3371 0.4427 0.1450 0.1112 0.4509 0.7948Columns 11 through 200.6443 0.3111 0.0855 0.0377 0.0305 0.0596 0.1734 0.9516 0.0326 0.25180.3786 0.9234 0.2625 0.8852 0.7441 0.6820 0.3909 0.9203 0.5612 0.29040.8116 0.4302 0.8010 0.9133 0.5000 0.0424 0.8314 0.0527 0.8819 0.61710.5328 0.1848 0.0292 0.7962 0.4799 0.0714 0.8034 0.7379 0.6692 0.26530.3507 0.9049 0.9289 0.0987 0.9047 0.5216 0.0605 0.2691 0.1904 0.82440.9390 0.9797 0.7303 0.2619 0.6099 0.0967 0.3993 0.42280.3689 0.98270.8759 0.4389 0.4886 0.3354 0.6177 0.8181 0.5269 0.5479 0.4607 0.73020.5502 0.1111 0.5785 0.6797 0.8594 0.8175 0.4168 0.9427 0.9816 0.3439 0.6225 0.2581 0.2373 0.1366 0.8055 0.7224 0.6569 0.4177 0.1564 0.5841 0.5870 0.4087 0.4588 0.7212 0.5767 0.1499 0.6280 0.9831 0.8555 0.1078 0.2077 0.5949 0.9631 0.1068 0.1829 0.6596 0.2920 0.3015 0.6448 0.9063 0.3012 0.2622 0.5468 0.6538 0.2399 0.5186 0.4317 0.7011 0.3763 0.8797 0.4709 0.6028 0.5211 0.4942 0.8865 0.9730 0.0155 0.6663 0.1909 0.8178 0.2305 0.7112 0.2316 0.7791 0.0287 0.6490 0.9841 0.5391 0.4283 0.2607 0.8443 0.2217 0.4889 0.7150 0.4899 0.8003 0.1672 0.6981 0.4820 0.5944 0.1948 0.1174 0.6241 0.9037 0.1679 0.4538 0.1062 0.6665 0.1206 0.0225 0.2259 0.2967 0.6791 0.8909 0.9787 0.4324 0.3724 0.1781 0.5895 0.4253 0.1707 0.3188 0.3955 0.3342 0.7127 0.8253 0.1981 0.1280 0.2262 0.3127 0.2277 0.4242 0.3674 0.6987 0.5005 0.0835 0.4897 0.9991 0.3846 0.1615 0.4357 0.5079 0.9880 0.1978 0.4711 0.1332 0.3395 0.1711 0.5830 0.1788Columns 21 through 300.4229 0.7788 0.2548 0.1759 0.6476 0.5822 0.4046 0.3477 0.8217 0.5144 0.0942 0.4235 0.2240 0.7218 0.6790 0.5407 0.4484 0.1500 0.4299 0.8843 0.5985 0.0908 0.6678 0.4735 0.6358 0.8699 0.3658 0.5861 0.8878 0.5880 0.4709 0.2665 0.8444 0.1527 0.9452 0.2648 0.7635 0.2621 0.3912 0.1548 0.6959 0.1537 0.3445 0.3411 0.2089 0.3181 0.6279 0.0445 0.7691 0.1999 0.6999 0.2810 0.7805 0.6074 0.7093 0.1192 0.7720 0.7549 0.3968 0.4070 0.6385 0.4401 0.6753 0.1917 0.2362 0.9398 0.9329 0.2428 0.8085 0.7487 0.0336 0.5271 0.0067 0.7384 0.1194 0.6456 0.9727 0.4424 0.7551 0.8256 0.0688 0.4574 0.6022 0.2428 0.6073 0.4795 0.1920 0.6878 0.3774 0.7900 0.3196 0.8754 0.3868 0.9174 0.4501 0.6393 0.1389 0.35920.2160 0.3185 0.5309 0.5181 0.9160 0.2691 0.4587 0.5447 0.6963 0.7363 0.7904 0.5341 0.6544 0.9436 0.0012 0.7655 0.6619 0.6473 0.0938 0.3947 0.9493 0.0900 0.4076 0.6377 0.4624 0.1887 0.7703 0.5439 0.5254 0.6834 0.3276 0.1117 0.8200 0.9577 0.4243 0.2875 0.3502 0.7210 0.5303 0.7040 0.6713 0.1363 0.7184 0.2407 0.4609 0.0911 0.6620 0.5225 0.8611 0.4423 0.4386 0.6787 0.9686 0.6761 0.7702 0.5762 0.4162 0.9937 0.4849 0.0196 0.8335 0.4952 0.5313 0.2891 0.3225 0.6834 0.8419 0.2187 0.3935 0.3309 0.7689 0.1897 0.3251 0.6718 0.7847 0.5466 0.8329 0.1058 0.6714 0.4243 0.1673 0.4950 0.1056 0.6951 0.4714 0.4257 0.2564 0.1097 0.7413 0.2703 0.8620 0.1476 0.6110 0.0680 0.0358 0.6444 0.6135 0.0636 0.5201 0.1971 0.9899 0.0550Columns 31 through 400.8507 0.7386 0.5523 0.1239 0.7378 0.5590 0.1781 0.8949 0.6311 0.6925 0.5606 0.5860 0.6299 0.4904 0.0634 0.8541 0.3596 0.0715 0.0899 0.5567 0.9296 0.2467 0.0320 0.8530 0.8604 0.3479 0.0567 0.2425 0.0809 0.3965 0.6967 0.6664 0.6147 0.8739 0.9344 0.4460 0.5219 0.0538 0.7772 0.0616 0.5828 0.0835 0.3624 0.2703 0.9844 0.0542 0.3358 0.4417 0.9051 0.78020.8154 0.6260 0.0495 0.2085 0.8589 0.1771 0.1757 0.0133 0.5338 0.33760.8790 0.6609 0.4896 0.5650 0.7856 0.6628 0.2089 0.8972 0.1092 0.60790.9889 0.7298 0.1925 0.6403 0.5134 0.3308 0.9052 0.1967 0.8258 0.74130.0005 0.8908 0.1231 0.4170 0.1776 0.8985 0.6754 0.0934 0.3381 0.10480.8654 0.9823 0.2055 0.2060 0.3986 0.1182 0.4685 0.3074 0.2940 0.12790.6126 0.7690 0.1465 0.9479 0.1339 0.9884 0.9121 0.45610.7463 0.54950.9900 0.5814 0.1891 0.0821 0.0309 0.5400 0.1040 0.1017 0.0103 0.48520.5277 0.9283 0.0427 0.1057 0.9391 0.7069 0.7455 0.9954 0.0484 0.89050.4795 0.5801 0.6352 0.1420 0.3013 0.9995 0.7363 0.3321 0.6679 0.79900.8013 0.0170 0.2819 0.1665 0.2955 0.2878 0.5619 0.2973 0.6035 0.73430.2278 0.1209 0.5386 0.6210 0.3329 0.4145 0.1842 0.0620 0.5261 0.05130.4981 0.8627 0.6952 0.5737 0.4671 0.4648 0.5972 0.2982 0.7297 0.07290.9009 0.4843 0.4991 0.0521 0.6482 0.7640 0.2999 0.0464 0.7073 0.08850.5747 0.8449 0.5358 0.9312 0.0252 0.8182 0.1341 0.5054 0.7814 0.79840.8452 0.2094 0.4452 0.7287 0.8422 0.1002 0.2126 0.7614 0.2880 0.9430Columns 41 through 500.6837 0.7894 0.1123 0.6733 0.0986 0.9879 0.5975 0.7593 0.8092 0.75190.1321 0.3677 0.7844 0.4296 0.1420 0.1704 0.3353 0.7406 0.7486 0.22870.7227 0.2060 0.2916 0.4517 0.1683 0.2578 0.2992 0.7437 0.1202 0.06420.1104 0.0867 0.6035 0.6099 0.1962 0.3968 0.4526 0.1059 0.5250 0.76730.1175 0.7719 0.9644 0.0594 0.3175 0.0740 0.4226 0.6816 0.3258 0.67120.6407 0.2057 0.4325 0.3158 0.3164 0.6841 0.3596 0.4633 0.5464 0.71520.3288 0.3883 0.6948 0.7727 0.2176 0.4024 0.5583 0.2122 0.3989 0.64210.6538 0.5518 0.7581 0.6964 0.2510 0.9828 0.7425 0.0985 0.4151 0.41900.7491 0.2290 0.4326 0.1253 0.8929 0.4022 0.4243 0.8236 0.1807 0.39080.5832 0.6419 0.6555 0.1302 0.7032 0.6207 0.4294 0.1750 0.2554 0.81610.7400 0.4845 0.1098 0.0924 0.5557 0.1544 0.1249 0.1636 0.0205 0.31740.2348 0.1518 0.9338 0.0078 0.1844 0.3813 0.0244 0.6660 0.9237 0.81450.7350 0.7819 0.1875 0.4231 0.2120 0.1611 0.2902 0.8944 0.6537 0.78910.9706 0.1006 0.2662 0.6556 0.0773 0.7581 0.3175 0.5166 0.9326 0.85230.8669 0.2941 0.7978 0.7229 0.9138 0.8711 0.6537 0.7027 0.1635 0.50560.0862 0.2374 0.4876 0.5312 0.7067 0.3508 0.9569 0.1536 0.9211 0.63570.3664 0.5309 0.7690 0.1088 0.5578 0.6855 0.9357 0.9535 0.7947 0.95090.3692 0.0915 0.3960 0.6318 0.3134 0.2941 0.4579 0.5409 0.5774 0.44400.6850 0.4053 0.2729 0.1265 0.1662 0.5306 0.2405 0.6797 0.4400 0.06000.5979 0.1048 0.0372 0.1343 0.6225 0.8324 0.7639 0.0366 0.2576 0.8667 2.参考课本综合例题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。

大连理工矩阵上机作业

第一题Lagrange插值函数function y=lagrange(x0,y0,x);n=length(x0);m=length(x);for i=1:mz=x(i);s=0.0;for k=1:np=1.0;for j=1:nif j~=kp=p*(z-x0(j))/(x0(k)-x0(j));endends=p*y0(k)+s;endy(i)=s;endx0=[1:10];y0=[67.052,68.008,69.803,72.024,73.400,72.063,74.669,74.487,74.065,76 .777];lagrange(x0,y0,17)ans=-1.9516e+12x=[1:0.1:10];x=x';plot(x0,y0,'r');hold onplot(x,y,'k');legend('原函数','拟合函数')拟合图像如下拟合函数出现了龙格现象，运用多项式进行插值拟合时，效果并不好，高次多项式会因为误差的不断累积，导致龙格现象的发生。

第二题function fun =nihe(n)m=[67.052*10^6,68.008*10^6,69.803*10^6,72.024*10^6,73.400*10^6,72.063 *10^6,74.669*10^6,74.487*10^6,74.065*10^6,76.777*10^6];w=[1,2,3,4,5,6,7,8,9,10];d1=0;d2=0;d3=0;y1=polyfit(m,w,1);y2=polyfit(m,w,2);y3=polyfit(m,w,3);y2=poly2sym(s2);y3=poly2sym(s3);y4=poly2sym(s4);f1=subs(y1,17);f2=subs(y2,17);f3=subs(y3,17);for p=1:10;d1=d1+(subs(y1,w(p))-m(p))^2;d2=d2+(subs(y2,w(p))-m(p))^2;d3=d3+(subs(y3,w(p))-m(p))^2;endd1=sqrt(d1);d2=sqrt(d2);d3=sqrt(d3);fun=[f1 f2 f3;d2 d3 d4];return;结果三次函数的均方误差最小，拟合的最好。

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。

大连理工大学庞丽萍最优化方法MATLAB程序

班级：优化1班授课老师：庞丽萍姓名：学号：第二章12.（1）用修正单纯形法求解下列LP问题:>>clear>>A=[121100;123010;215001];[m,n]=size(A);b=[10;15;20];r=[-1-2-31];c=[-1-2-31];bs=[3:3];nbs=[1:4];a1=A(:,3);T=A(:,bs);a2=inv(T)*a1;b=inv(T)*b;A=[eye(m),a2];B=eye(m);xb=B\b;cb=c(bs);cn=c(nbs);con=1;M=zeros(1);while conM=M+1;t=cb/B;r=c-t*A;if all(r>=0)x(bs)=xb;x(nbs)=0;fx=cb*xb;disp(['当前解是最优解，minz=',num2str(fx)])disp('对应的最优解为,x=')disp(x)breakendrnbs=r(nbs);kk=find(rnbs==min(rnbs));k=kk(1);Anbs=A(:,nbs);yik=B\Anbs(:,k);xb=B\b;%yi0if all(yik<=0)disp('此LP问题无有限的最优解，计算结束',x)disp(xb)breakelsei=find(yik>0);w=abs(xb(i,1)./yik(i,1));l=find(w==min(w));rr=min(l);yrrk=yik(rr,1);Abs=A(:,bs);D=Anbs(:,k);Anbs(:,k)=Abs(:,rr);Abs(:,rr)=D;F=bs(rr);bs(rr)=nbs(k);nbs(k)=F;AA=[Anbs,Abs];EE=eye(m);EE(:,rr)=-yik./yrrk;Errk=EE;Errk(rr,rr)=1/yrrk;BB=Errk/B;B=inv(BB);cb=c(:,bs);xb=Errk*xb;x(bs)=xb;x(nbs)=0;fx=cb*xb;endif M>=1000disp('此问题无有限最优解')breakendend%结果当前解是最优解，minz=-15对应的最优解为,x=2.5000 2.5000 2.50000第三章30题DFP算法求函数极小点的计算程序function[x,val,k]=dfp(fun,gfun,x0)%功能:用DFP算法求解无约束问题:minf(x)%输入:x0是初始点,fun,gfun分别是目标函数及其梯度%输出:x,val分别是近似最优点和最优值,k是迭代次数.maxk=1e5;%给出最大迭代次数rho=0.55;sigma=0.4;epsilon=1e-5;k=0;n=length(x0);Hk=inv(feval('Hess',x0));%Hk=eye(n);while(k<maxk)gk=feval(gfun,x0);%计算梯度if(norm(gk)<epsilon),break;end%检验终止准则dk=-Hk*gk;%解方程组,计算搜索方向m=0;mk=0;while(m<20)%用Armijo搜索求步长if(feval(fun,x0+rho^m*dk)<feval(fun,x0)+sigma*rho^m*gk’*dk)mk=m;break;endm=m+1;end%DFP校正x=x0+rho^mk*dk;sk=x-x0;yk=feval(gfun,x)-gk;if(sk'*yk>0)Hk=Hk-(Hk*yk*yk'*Hk)/(yk'*Hk*yk)+(sk*sk')/(sk'*yk);endk=k+1;x0=x;endval=feval(fun,x0);%习题26的程序调用方式及结果：function y=fun(x)%UNTITLED Summary of this function goes here%Detailed explanation goes herey=(x(1)-1)^2+5*(x2-x(1)^2)^2endfunction y=gfun(x)%UNTITLED Summary of this function goes here%Detailed explanation goes herey=[diff(y,x1)diff(y,x2)]endx0=[20]’;[x,val,k]=dfp(fun,gfun,x0)%结果x=1.000001.00000val=k=6%习题27的程序调用方式及结果：function y=fun(x)%UNTITLED Summary of this function goes here %Detailed explanation goes herey=x1+2*x（2）^2+exp(x(1)^2+x(2)^2)endfunction y=gfun(x)%UNTITLED Summary of this function goes here %Detailed explanation goes herey=[diff(y,x1)diff(y,x2)]endx0=[10]’;[x,val,k]=dfp(fun,gfun,x0)%结果x=-0.419360val=0.77291k=536题编写Hooke-Jeeves方法求函数极小点的计算程序。

大连理工大学庞丽萍最优化方法MATLAB程序

优化方法MATLAB编程——大连理工大学

优化方法上机大作业学院：姓名：学号：指导老师：肖现涛第一题源程序如下：function zy_x = di1ti(x)%di1ti是用来求解优化作业第一题的函数。

x0=x; yimuxulong=0.000001;g0=g(x0);s0=-g0;A=2*ones(100,100);k=0;while k<100lanmed=-(g0)'*s0/(s0'*A*s0);x=x0+lanmed*s0;g=g(x);k=k+1;if norm(g)<yimuxulongzy_x=x;fprintf('After %d iterations,obtain the optimal solution.\n \n The optimal solution is \n %f.\n\nThe optimal "x" is "ans".',k,f(x) )break;endmiu=norm(g)^2/norm(g0)^2;s=-g+miu*s0;g0=g; s0=s;x0=x;endfunction f=f(x)f=(x'*ones(100,1))^2-x'*ones(100,1);function g=g(x)g=(2*x'*ones(100,1))*ones(100,1)-ones(100,1);代入x0,运行结果如下：>> x=zeros(100,1);>> di1ti(x)After 1 iterations,obtain the optimal solution.The optimal solution is-0.250000.The optimal "x" is "ans".ans =0.005*ones(100,1).第二题1.最速下降法。

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):。

大连理工大学--优化作业----程序共9页文档

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);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。

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：。

大连理工大学-2015年矩阵上机编程作业

cout<<endl; } } }
} //Ax=b 的解答，A 为上三角矩阵 void SolveOne(double a[9][10],int m,int n){
int j=n-2; double answer[9]; for (int s=m-1;s>=0;s--){
answer[j]=a[s][n-1]/a[s][j]; for(int i=0;i<=s-1;i++){
} } }
} // 发现绝对值最大的元素所在一行和当前的一行做交换 void change(double a[9][10],int m,int n,int nowi,int nowj){
int record=nowi;//记录该和哪一行做交换 bool flag=true; //标志是否需要做交换 double max=a[nowi][nowj]; for(int i=nowi+1;i<m-1;i++){
}
//Gauss 消元解法 //先转化为上三角矩阵 void Gauss(double a[9][10],int m,int n){
double r[9]; //r 为每次的倍数 int k=0; for (int nowi=0,nowj=0;nowi<m-1 && nowj<n-1;nowi++,nowj++){
1. 设
, 其精确值为
.
(1) 编制按从大到小的顺序
, 计算的通
用程序 (2) 编制按从小到大的顺序
, 计算
的通用程序
(3) 按两种顺序分别计算
并指出有效位数（编制程序时用单精度）

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

优化方法上机作业--2013

优化方法上机作业--2013第一题（牛顿法和不精确一维搜索）牛顿法：syms x1 x2;f=(x2-(x1)*(x1))^2+(1-x1)^2;v=[x1,x2];df=jacobian(f,v);df=df.';G=jacobian(df,v);epson=1e-12;x0=[0,1]';g1=subs(df,{x1,x2},{x0(1,1),x0(2,1)});G1=subs(G,{x1,x 2},{x 0(1,1),x0(2,1)});k=0;mul_count=0;sum_count=0;mul_count=mul_count+12;sum_count=sum_count+6;while(norm(g1)>epson)p=-G1\g1;x0=x0+p;g1=subs(df,{x1,x2},{x0(1,1),x0(2,1)});G1=subs(G,{x1,x2},{x0(1,1),x0(2,1)});k=k+1;mul_count=mul_count+16;sum_count=sum_count+11;end;kx0mul_countsum_count运行结果：k =9x0 =11mul_count =156sum_count =105不精确一维搜索：fun1.m文件function f=fun1(x)f=(x(1)-1)^2+(x(2)-x(1)^2)^2;gfun1.m文件function gf = gfun1(x)gf=[2*x(1)*(x(1)^2-x(2))+2*(x(1)-1),2*(x(2)-x(1)^2)]';wolfepowell.m文件function [k,m,opt,x]=wolfepowell(xk,sk)max=1000;c1=0.1; c2=0.6;a=0;b=inf;dk=0.2;m=0;while(m<=max)if (fun1(xk)-fun1(xk+sk*dk)<-c1*dk*gfun1(xk)'*sk)b=dk;dk=(dk+a)/2;elseif(fun1(xk)-fun1(xk+sk*dk)>=-c1*dk*gfun1(xk)'*sk)&&(gfun1(xk+sk*dk)'*skif(gd>=0.0)d=-g;endendif(norm(g)<="" p="">m=0; mk=0;while(m<20)if(feval(fun,x0+rho^m*d)<feval(fun,x0)+sigma*rho^m*g'*d) < p="">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);fun.m文件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); gfun.m文件function gf=gfun(x)gf=[2*x(1)-2*x(2)+2, -2*x(1)+4*x(2)+3,2*x(3)-x(2)-1,2*x(4)]';运行结果：>>x0=[0 0 0 0]';[x,val,k]=frcg('fun','gfun',x0)x =-3.4064-2.6515-0.7032val =-7.8338k =5000第三题（BFGS法、最速下降法和牛顿法）fun3.m文件function f=fun3(x)f=(x(1)-1)^2+5*(x(2)-x(1)^2)^2;gfun3.m文件function gf = gfun3(x)gf=[20*x(1)*(x(1)^2-x(2))+2*(x(1)-1),10*(x(2)-x(1)^2)]'; Hessen.m文件function Hess=Hessen(x)Hess=[60*x(1)^2-20*x(2)+2,(-20)*x(1);(-20)*x(1),10];Steepest.m文件function [m,opt,x] = Steepest(xk)m=0;Eps=1.0e-4;max=1000;while(m<max)< p="">g=gfun3(xk);sk=-g;dk=wolfepowell(xk,sk);x=xk+dk*sk;g0=gfun3(x);if(norm(g0)<=Eps)break;elsexk=x;endm=m+1;endmk=m;x=xk;opt=fun3(xk);newton.m文件function [x,opt,mk] = newton(xk)m=0;Eps=1e-4;max=100;H=[1,0;0,1]; while(m<max)< p="">g=gfun3(xk);M=Hess(xk);H=inv(M);sk=-H*g;dk=wolfepowell(xk,sk);x=xk+dk*sk;g0=gfun3(x);if(norm(g0)<=Eps)break;elsexk=x;endm=m+1;endmk=m;x=xk;opt=fun3(xk);bfgs.m文件function [x,opt,mk] = bfgs(xk)m=0;Eps=1e-4;max=100;H=[1,0;0,1]; while(m<max)< p="">g=gfun3(xk);sk=-H*g;dk=wolfepowell(xk,sk);x=xk+dk*sk;g0=gfun3(x);if(norm(g0)<=Eps)break;elseH=H+((1+((g0-g)'*H*(g0-g))/((x-xk)'*(g0-g)))*(x-xk)*(x-xk)'-H*(g0-g)*(x-xk)'-(x-x k)*(g0-g)'*H)/((x-xk)'*(g0-g));xk=x;endm=m+1;endmk=m;x=xk;opt=fun3(xk);wolfepowell.m文件function [k,m,opt,x]=wolfepowell(xk,sk)max=1000;c1=0.1; c2=0.6;a=0;b=inf;dk=0.2;m=0;while(m<=max)if (fun3(xk)-fun3(xk+sk*dk)<-c1*dk*gfun3(xk)'*sk)b=dk;dk=(dk+a)/2;elseif(fun3(xk)-fun3(xk+sk*dk)>=-c1*dk*gfun3(xk)'*sk)&&(gfun3(xk+sk*dk)'*skx =1.00001.0000opt =3.2601e-10mk =12最速下降法运行结果：>>xk=[2,0]';[x,opt,mk] = Steepest(xk)x =0.99990.9998opt =8.2108e-09mk =354BFGS法运行结果：>>xk=[2,0]';[x,opt,mk] = bfgs(xk)x =0.99990.9999opt =4.7725e-09mk =16(0.9999, 0.9998) 最速下降法（迭代次数354）即得数值最优点为x= (1.0000, 1.0000) 牛顿法（迭代次数12）(0.9999, 0.9999) BFGS公式（迭代次数16）最优解近似为0第四题（乘子法）multphr.m文件function[x,mu,lambda,output]=multphr(fun,hf,gf,dfun,dhf,dgf,x0) maxk=500;sigma=2.0;eta=2.0; theta=0.8;k=0; ink=0;epsilon=1e-5;x=x0; he=feval(hf,x); gi=feval(gf,x);n=length(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,~,ik]=bfgs('mpsi','dmpsi',x0,fun,hf,gf,dfun,dhf,dgf,mu,lam bda,sigma);ink=ink+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),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(0.0,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;mpsi.m文件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;endpsi=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;endpsi=psi+s2/(2.0*sigma);h1.m文件function he=h1(x)he=-x(1)^2-x(2)^2+25.0;f1.m文件function f=f1(x)f=4*x(1)-x(2)^2-12;g1.m文件function gi=g1(x)gi=10*x(1)-x(1)^2+10*x(2)-x(2)^2-34;h1.m文件function dhe = dh1(x)dhe = [-1*x(1), -1*x(2)]';dg1.m文件function dgi = dg1(x)dgi = [10-2*x(1), 10-2*x(2)]';df1.m文件function g=df1(x)g = [4, -2.0*x(2)]';bfgs.m文件function [x,val,k]=bfgs(fun,gfun,x0,varargin) maxk=500;rho=0.55; sigma1=0.4; epsilon1=1e-5;k=0; n=length(x0);Bk=eye(n);while(k<maxk)< p="">gk=feval(gfun,x0,varargin{:});if(norm(gk)<="" p="">dk=-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+sigma1*rho^m*gk'*dk)< p="">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{:});dmpsi.m文件functiondpsi=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);endfor(i=1:m)dpsi=dpsi+(sigma*gi(i)-lambda(i))*dgi(:,i);end运行结果：>>x0=[1,1]';[x,mu,lambda,output]=multphr('f1','h1','g1','df1','dh1','dg1', x0) x =1.00134.8987mu =2.0312lambda =0.7545output =fval: -31.9923iter: 5inner_iter: 58bta: 4.3187e-07第五题（有效集法）qpact.m文件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,:)]; endendgk=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)< p=""> 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;qsubp.m文件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 = zeros(m,1);endcallqpact.m文件function 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) 运行结果：>>callqpactx =1.40001.7000lambda =0.8000exitflag =output =fval: -6.4500iter: 7第七题fun文件function f=fun(x)f=abs(x(1))+abs(x(2))+abs(x(3))+abs(x(4));主程序x0=[1;1;1;1];A=[]; b=[];Aeq=[1 0 0 0.5;0 1 0.2 0.3;0 0.1 1 0.2];beq=[-1;0.2;1]; VLB=[]; VUB=[];[x,fval]=fmincon('fun',x0,A,b,Aeq,beq,VLB,VUB)运行结果x =-1.0000-0.00001.00000.0000 fval =2.0000</tm)<></oldf+sigma1*rho^m*gk'*dk)<></maxk)<></maxk)<></max)<></max)<></max)<></feval(fun,x0)+sigma*rho^m*g'*d)<></maxk)<>。

大连理工大学数据结构(一)上机作业答案——张老师

#define OVERFLOW -2
typedef int ElemType;
typedef int Status;
#define LIST_INIT_SIZE 100
#define LISTTINCREMENT 10
typedef struct{
ElemType *elem;
int length;
for(i=0;i<n;i++)
{
printf("input elem:");
scanf("%d",&L.elem[i]);
}
}
//输出顺序表中的元素
void ListOutput_Sq(SqList L){
int i,n;
n=L.length;
for(i=0;i<n;i++)
printf("%2d",L.elem[i]);
else
return -2;
}//优先级比较
void transform(char suffix[], char exp[])
{
char *p, ch, c;
p = exp;
ch = *p;
SqStack S;
InitStack (S);
Push (S, '#');
while(!StackEmpty(S))
}
}
}
void DeleteList(LinkList &L,ElemType mink,ElemType maxk){
LNode *p=L,*q;
while(p->next&&p->next->data<=mink)

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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大连理工大学优化方法上机作业-标准化文件发布号：（9456-EUATWK-MWUB-WUNN-INNUL-DDQTY-KII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=0.5; sigma=0.2;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=0.5; sigma=0.4;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 =0.99990.9998val =1.2390e-008%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 =1.00001.0000miny =4.9304e-030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=0.55;sigma=0.4;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 =1.00001.0000val =2.2005e-011%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=0.6;sigma=0.4;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 =1.00011.0002val =7.2372e-009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)+0.5*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=0.5;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'*dkmk=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=-31.4003x=[1.0984;4.8779]val_min =-31.4003上机大作业（三）A=[1 1;-1 0;0 -1];n=2;b=[1;0;0];G=[0.5 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): -2.40.40000.6000A=[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): -5.40.20000.00001.600011。

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