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

Abstract

The radius of robust feasibility of a convex program with uncertain constraints gives a value for the maximal 'size' of an uncertainty set under which robust feasibility can be guaranteed. This paper provides an upper bound for the radius for convex programs with uncertain convex polynomial constraints and exact formulas for convex programs with SOS-convex polynomial constraints (or convex quadratic constraints) under affine data uncertainty. These exact formulas allow the radius to be computed by commonly available software.

A note on the radius of robust feasibility for uncertain convex programs

“…Then, by using the exact calculating formula, we obtain a necessary and sufficient condition for robust feasibility for uncertain convex programs to be positive. Moreover, it is shown that our results can be refined in the special cases of [10,[16][17][18].…”

“…Therefore, the notion of the epigraphical set was inspired by the concept of hypographical set for linear semi-infinite program of [14]. Following [10,[16][17][18], we define the concept of radius of the robust feasibility for problem (RP α ) as follows.…”

“…Goberna et al [17] first established the next lemma to present the bounds for the radius of robust feasibility.…”

“…In [16], Goberna et al employed a new proof way, which is different from the one of [10], to obtain the exact formula for the radius of robust multi-objective programs. Goberna et al [17] extended the main results of [16] to robust convex programs. In 2017, Chuong and Jeyakumar [18] observed that the formula are valid only for programs with a ball uncertainty set, and then study the radius of robust feasibility for uncertain linear programs under a general convex and compact uncertainty.…”

“…Motivated by the work reported in [10,[16][17][18], this paper aims to establish some results of the radius of robust feasibility for uncertain convex programs under a general convex and compact uncertainty. The distinguishing feature of our work lies in the use of convex and compact uncertainty set which includes the ellipsoidal, ball, polytope and box uncertainty sets and so on.…”