Semi-definite Programming Relaxations Method For Convex Semi-infinite Polynomial Programming Problems

Semi-infinite programming is a classic optimization problem,which plays an important role in many fields.In recent years,semi-infinite programming has attracted great attention and attention from scholars internationally,which has achieved rapid development.Consequently,rich results have been achieved in theory,algorithms and applications.In this paper,we mainly study a class of convex semi-infinite programming problems de-fined on semi-infinite systems of polynomial inequalities,where the objective function and the functions involved in the constraints are all SOS convex polynomial functions,and the index set in the constraints is several types of simple basic semi-algebraic sets.According to existing work,such a problem can be reformulated to a conic optimization problem.After that,the Putinar's Positivstellensatz can be used to describe the inner approximation of the set,which composed of nonnegative polynomials on the index set.In addition,we can also use the representation of non-negative polynomials on the semi-algebraic set to further construct an inner approximate semidefinite representation of the feasible set.Under some assumptions,the semidefinite repre-sentation can converge to the original set.The key work of this paper consists of two parts.One is to derive the lower bound of this kind of convex semi-infinite programming problem.And the other is to obtain the outer ap-proximate semidefinite representation of the feasible set,which is composed of SOS convex polynomials.In order to derive the lower bounds of the original problem,we consider the intro-duction of measure analysis and sum of squares in the conic constraints.And we construct the dual set of the sum of squares set as the outer approximation of the set,which formed by nonneg-ative polynomials on the index set.In addition,we can also use the representation of nonnegative polynomials on the semi-algebraic set to further construct an outer approximate semidefinite rep-resentation of the feasible set.Under some assumptions,the semidefinite representation can can converge to the original set.In addition,we will also use MATLAB programming software to carry out numerical experiments to analyze the efficiency of the SDP relaxation problems when the index set in the constraint condition is several types of simple semi-algebraic sets.