Some Researches On The Problem Of Linear Quadratic Semi-definite Programming

In this paper,we discuss the feature and solving algorithms for the Linear Quadratic Semi-definite Programming(L-QSDP).This paper mainly consists of three sections.In the first section,we establish the duality theory and the optimality conditions in the problem of L-QSDP,and study the primal dual interior point method for this problem.Meanwhile,we give the proof of the unique solution based on the NT search direction.Furthermore,we make the MATLAB code for experiment when the number of dimension n=3,and give the numerical simulation in MATLAB 7.01, which verify its feasibility.Finally,we discuss the relation between the quadratic semi-definite programming and the semi-definite least squares problem,and then provide the conversion relationship between them under some conditions.In the second section,we extend the definition of the semi-definite least squares problem(SDLS),and propose the semi-definite least squares problem with bounded variables(BV-SDLS).At the same time,we give the projection of symmetric matrix onto the closed convex set,which is composed of the boundary constrains.We also study the solving algorithm for(BV-SDLS)based on this projection,i.e.the projected quasi-newton algorithm,and give the algorithm frame for this problem.In the end,we make an experiment in MATLAB 7.01,and compare numerical results with that of the primal dual interior point method.Numerical results show that the extended approach is feasible and validity.In the third section,we further discuss the extension of(BV-SDLS),and propose the explicit projective formula,in certain particular circumstances.Moreover,we also study its solving algorithm.