Robust linear semi-infinite programming duality under uncertainty

Abstract

In this paper, we propose a duality theory for semi-in…nite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-in…nite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then show that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and su¢ cient for robust duality in the sense that robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with a¢ nely parameterized data uncertainty, we establish that robust duality is easily satis…ed under a Slater type constraint quali…cation. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-in…nite linear inequalities.

Robust Solutions of MultiObjective Linear Semi-Infinite Programs under Constraint Data Uncertainty

“…As a corollary, we obtain a version of the robust duality theorem which was given in [10] for a single-objective linear semi-infinite programming problem under data uncertainty using a local constraint qualification. Corollary 4.3.…”

“…Recently, single-objective linear semi-infinite programming problems under constraint data uncertainty were studied in [10].…”

“…In the same vein as in [10], the multi-objective linear problem (P ) in the face of data uncertainty in the constraints can be captured by a parameterized multi-objective linear problem of the form (1.2) (P u ) V-min x∈R n c 1 x, . .…”

“…In this section, we now develop a suitable robust duality framework for the multi-objective linear semi-infinite programming problem with uncertain constraints. Related details for scalar optimization problems can be found in [1,5,10,14,15].…”

“…c − t∈T y t v t = 0 n , which is the optimistic counterpart of the usual Lagrangian dual of a semi-infinite programming problem introduced in [10].…”