Existence of Robust Optimal Solutions for Uncertain Convex Polynomial Optimization Problems

The paper is devoted to the existence of robust optimal solutions for convex polynomial optimization problems in the face of data uncertainty both in the objective function and the constraints. Employing adequate tools of real semi-algebraic geometry, we first introduce the concept called robust tangency variety and investigate its properties. Then under the robust Mangasarian-Fromovitz constraint qualification, we establish connections between robust Palais-Smale condition, robust weak Palais-Smale condition, robust M-tameness, and condition (A) formulated for the restriction of the objective function on the constraint set and uncertain set associated with the objective function. Finally, based on obtained relationships, we derive some sufficient conditions for the existence of robust optimal solutions of the considered problem.