Comprehensive survey on convex analysis in robust optimization

Abstract

This paper presents a comprehensive survey on Convex Analysis (CA) in Robust Optimization (RO). Since RO is a class of continuous optimization problem which involved uncertain parameter thus convex analysis is needed to guarantee that the uncertain set is compact set, i.e., closed and bounded. Furthermore, in RO, to handle the uncertainty, the Robust Counterpart Methodology is employed to guarantee that the Robust Counterpart (RC) of the uncertain optimization problem will be ended up in one of the computationally tractable problems, whether it is a linear optimization, conic quadratic or semidefinite optimization problems. The importance of CA in RO lies in a fact that the robustness can be represented by a convex hull, i.e., the set which is the smallest convex set such that the uncertainty set is included in. This convex hull can replaces the uncertainty set. This can be done since determining the solution feasibility with respect to an uncertainty set is equivalent to taking the constraint left hand side supremum over the uncertainty set. If the uncertainty set can be replaced by a convex hull, this yields that the optimal solution remains the same. Thus, this paper discuss the recent researches, the most cited paper, the top journal and publishers also the bibliometric analysis on the topic of CA in RO.