Applications Of Lasserre Relaxation Method In Quadratic Programming Problems

Polynomial optimization problem (POP) is a class of important optimization problem. It has been applied in many areas, such as signal processing, system control theory, as a kind of mathematical model.Therefore, how to solve the POP is an interesting problem. Recently, J.B. Lasserre proposed a method which is focused on finding out the global optimal value of POP. The method, for short Lasserre relaxation method, has been attracted attentions of many optimization experts.There are two main contents in this paper, one of which is to apply software package GloptiPoly, that was based on Lasserre relaxation method and was developed by Henrion and Lasserre, to solve quadratic programming and to test its numerical performances, other of which is to construct a trust domain method whose subproblem is solved by Lasserre relaxation method and to test its numerical performances.In Chapter 1, we briefly describe polynomial optimization problem and Lasserre relaxation method.In Chapter 2, we apply Lasserre relaxation method to solve various quadratic programming prob-lems, including unconstrained quadratic programming, linear constrained quadratic programming and quadratic constrained quadratic programming with stochastic data. Numerical results illustrate that GloptiPoly can efficiently solve most of stochastic numerical examples with middle and small scales, including almost singular examples.In Chapter 3, firstly, trust domain algorithms aiming at unconstrained nonlinear programming and linear constrained nonlinear programming are put forward. Secondly, convergence properties of these trust domain algorithms are given. Finally, numerical experiments show that the algorithms are effective.At last, simple summaries and expectations of the paper are made.