A new class of alternative theorems for SOS-convex inequalities and robust optimization

Abstract

In this paper, we present a new class of alternative theorems for SOS-convex inequality systems without any qualifications. This class of theorems provides an alternative equations in terms of sums of squares to the solvability of the given inequality system. A strong separation theorem for convex sets, described by convex polynomial inequalities, plays a key role in establishing the class of alternative theorems. Consequently, we show that the optimal values of various classes of robust convex optimization problems are equal to the optimal values of related semidefinite programming problems (SDPs) and so, the value of the robust problem can be found by solving a single SDP. The class of problems includes programs with SOS-convex polynomials under data uncertainty in the objective function such as uncertain quadratically constrained quadratic programs. The SOS-convexity is a computationally tractable relaxation of convexity for a real polynomial. We also provide an application of our theorem of the alternative to a multi-objective convex optimization under data uncertainty.

SOS-Convex Semialgebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

“…(3) In Section 5, we briefly outline how our results can be applied to show that robust SOSconvex optimization problems under restricted spectrahedron data uncertainty enjoy exact semi-definite programming relaxations. This extends the existing result for restricted ellipsoidal data uncertainty established in [13] and answers the open questions left in [13] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.…”

“…In this section, we briefly outline how our results can be applied to the area of robust optimization [4] (for some recent development see [8,9,12,13]). Consider the following robust SOS-convex optimization problem…”

“…An appealing feature of an SOSconvex polynomial is that deciding whether a polynomial is SOS-convex or not can be equivalently rewritten as a feasibility problem of a semi-definite programming problem (SDP) which can be validated efficiently. It has also been recently shown that for an SOS-convex optimization problems, its optimal value and optimal solution can be found by solving a single semi-definite programming problem [19] (see also [13,14]). On the other hand, many modern applications of optimization to the area of statistics, machine learning, signal processing and image processing often result in structured nonsmooth convex optimization problems [6].…”

A bilevel Farkas lemma to characterizing global solutions of a class of bilevel polynomial programs

“…When it comes to deriving mathematical principles of optimality or duality for optimization problems, there is no better mathematical tool than the Farkas lemma or its generalizations [2,10,9,12,14,16,22]. The hundred years old celebrated Farkas lemma is a fundamental result for systems of linear inequalities.…”

Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints

“…T (x − y) is a sum-of-squares polynomial in (x, y) on R n × R n (see [1,18]) or, equivalently, if its Hessian matrix ∇ 2 f is SOS matrix polynomial (i.e.,…”