Stability and Genericity for Semi-algebraic Compact Programs

Abstract

In this paper, we consider the class of polynomial optimization problems over semi-algebraic compact sets, in which the objective functions are perturbed, while the constraint functions are kept fixed. Under certain assumptions, we establish some stability properties of the global solution map, of the Karush---Kuhn---Tucker set-valued map, and of the optimal value function for all problems in the class. It is shown that, for almost every problem in the class, there is a unique optimal solution for which the global quadratic growth condition and the strong second-order sufficient conditions hold. Furthermore, under local perturbations to the objective function, the optimal solution and the optimal value function (respectively, the Karush---Kuhn---Tucker set-valued map) vary smoothly (respectively, continuously) and the set of active constraint indices is constant. As a nice consequence, for almost every polynomial optimization problem, there is a unique optimal solution, which can be approximated arbitrarily closely by solving a sequence of semi-definite programs.