不等式约束最优化问题的可行SQP下降算法及其收敛性_英文_

Chin.Quart.J.of Math.

2009,24(3):469—474

Feasible SQP Descent Method for Inequality Constrained Optimization Problems

and Its Convergence

ZHANG He-ping1,YE Liu-qing2

(1.Department of Mathematics,Luohe Vocational Technology College,Luohe462000,China;2.Depart-ment of Mathematics,Jiaozuo Teachers College,Jiaozuo454002,China)

Abstract:In this paper,the new SQP feasible descent algorithm for nonlinear constrained

optimization problems presented,and under weaker conditions of relative,we proofed the

new method still possesses global convergence and its strong convergence.The numerical

results illustrate that the new methods are valid.

Key words:nonlinearly constrained optimization;SQP;the generalized projection;line

search;global convergence;strong convergence.

2000MR Subject Classification:90C30,49M37

CLC number:O221.2Document code:A

Article ID:1002–0462(2009)03–0469–06

§1.Introduction and Preparation Concept Nonlinear inequality constrained optimization problems

(P)min f0(x);

s.t.f j(x)≤0,j∈I={1,2,···,m}

During the research of method for nonlinear inequality constrained optimization,Due to the rapid convergence-type methods(such as sequential quadratic programming,sequential quadratic programming,Jane recorded as SQP)can calculate the solution to meet the optimization in the shortest possible time and with less the amount of calculation under a certain accuracy solution,

Received date:2009-03-10

Foundation item:Supported by the NNSF of China(10231060);Supported by the Soft Science Foundation of Henan Province(082400430820)

Biographies:ZHANG He-ping(1965-),male,native of Hebi,Henan,an associate professor of Luohe Vo-cational Technology College,M.S.D.,engages in nonlinear programming;YE Liu-qing(1965-),male,native of Runan,Henan,a professor of Jiaozuo Teachers College,M.S.D.,engages in nonlinear programming.

470CHINESE QUARTERLY JOURNAL OF MATHEMATICS Vol.24

which is widely used in economic planning.Production management,engineering design,power distribution,oil exploration,government decision-making and many other important areas.

SQP is a kind of effective method to study rapid convergence of the algorithm,the earliest research work was done by Wilson in his doctoral thesis[1]the Solver algorithm in1963.SQP early iteration point of the algorithm and approximate solutions can not satisfy the feasibility request,and the most recent SQP algorithm uses penalty function as an effective function, requesting the initial point must be feasible,and be a newflaw,in order to satify the require-ments of many practical issues on the approximate solution and even the strict conditions of the feasibility of point iteration,Panier-Tits[2]firstly adopted the technique of each of the three iterative solution of quadratic programming in1987,and established a feasible method both the initial point and iteration point of problem(P),later the idea has got further research and development,China’s well-known experts and scholars such as Gao Ziyou,Jian Jinbao,Li Zongbao,Xue Shengjia,have got a wealth of research results[3−8].

In the issue of(P),we make,f j(x)=0,(j∈I∪E={m+1,m+2,···,n}).Set x∈R n, f0(x),f j(x):R n→R(j∈I∪E)to be continuously differentiable function,marks feasible set

X={x∈R n|f i(x)≤0,i∈I},f j(x)=0,j∈E,

a positive constraints set of inequality is I(x)={j∈I|f j(x)=0},the Lagrange function is L(x,u)=f0(x)+

j∈L

u j f j(x),j∈L=I∪E.

§2.Issue and Hypothesis

Let’s consider the inequality constrained optimization problem(P).Marking problem(P)-feasible,gradient,the array are as follows:

X={x|f j(x)≤0,j∈I},g j(x)= f j(x),H j(x)= 2f j(x),j=0,1,···,m.

On issue(P)we make the assumption(H1)as follows:

Assumption(H1)(1)Function f0(x),f j(x)(j∈I)are at least twice continuous differen-tiability;

(2)∀x∈X,Vector Group{g j(x),j∈I(x)}are linearly independent,I(x)={j∈I|f j(x)= 0}is one positive constraint set.

The problem(P)’s Lagrange function and the Hessian matrix as:

L(x,u)=f j(x)+

j∈I u j f j(x),∇2xx L(x,u)=H0(x)+

j∈I

u j H j(x).

For feasible solution x∈X and its matrix B x,we consider the following quadratic programming

(p )min g0(x)T d+1

2

d T B x d;

s.t.f j(x)+g j(x)T d≤0,j∈I.