Strong solvability of restricted interval systems and its applications in quadratic and geometric programming

Abstract

We consider interval systems of linear equations and inequalities with a restriction to some a priori given set. We focus on a characterization of strong solvability, that is, solvability for each realization of interval values, and we compare this with an existence of a strong solution defined analogously. The motivation comes from the area of interval-valued optimization problems, where strong solvability means guaranteed feasibility of any realization of the problem. Strong solvability with strict inequalities implies the robust Slater condition, which ensures that standard optimality conditions can be used. We apply the issues particularly in two optimization classes, convex quadratic programming with quadratic constraints and posynomial geometric programming. For the former, we also utilize the presented result to improve a characterization of the worst case optimal value. Eventually, we state several open problems that emerged while deriving the results.