Characterizations of Robust and Stable Duality for Linearly Perturbed Uncertain Optimization Problems

Abstract

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.

Primal–Dual Optimization Conditions for the Robust Sum of Functions with Applications

“…Adopting the robust optimization approach under uncertainty (as in [4], [6], [7], [15], etc.) we have shown in [8] that (RP x * ) may be interpreted as the robust optimization counterpart of some uncertain optimization problem and (RD x * ) as its optimistic dual.…”

Duality for the Robust Sum of Functions

“…Given a locally convex Hausdorff topological vector space X and an infinite family (f i ) i∈I ⊂ (R ∞ ) X , where R ∞ := R∪ {+∞} , of objective proper functions, we are concerned with the uncertain problem of minimizing a finite but unknown sum of the objective functions f i . Adopting the robust optimization approach under uncertainty (see [3], [6], [7], [11]), and taking the set F (I) of non-empty finite subsets of I as uncertainty set, the robust counterpart of this uncertain problem is…”

Convexity and closedness in stable robust duality

“…On the other hand, both types of duality are simultaneously studied in [2] and [5], where characterizations of robust and robust stable strong duality are given for non-convex and/or convex robust problems. It is worth mentioning that among the mentioned papers, some provide perturbational schemes covering optimization problems with uncertain constraints and linear perturbations of the objective functions, such as [3], [5], [14].…”

“…In this short paper, following [3], we consider a given family fF u : u 2 U g of perturbation functions, where the index set U is called the uncertainty set of the family, F u : X Y u ! R 1 := R [ f+1g; and the decision space X and the parameter spaces Y u ; u 2 U; are locally convex Hausdor¤ topological vector spaces.…”

“…R 1 := R [ f+1g; and the decision space X and the parameter spaces Y u ; u 2 U; are locally convex Hausdor¤ topological vector spaces. In contrast with the "classical" robust duality scheme (as in [14]), where a unique parameter space Y is considered, in our model the parameter space Y u depends on u (this dependence is illustrated with a realistic production planning problem in [3,Section 2,Case 3]). Here, we specialize the totally general results obtained in [3] to problems satisfying certain convexity and closedness properties.…”