Adjustable robust multiobjective linear optimization: Pareto optimal solutions via conic programming

In this paper we study two-stage affinely adjustable robust multi-objective optimization problems. We show how (weak) Pareto optimal solutions of these robust multi-objective problems can be found by solving conic linear programming problems. We do this by first deriving numerically verifiable conditions that characterize (weak) Pareto optimal solutions of affinely adjustable robust multi-objective programs under a spectrahedron uncertainty set. The uncertainty set covers most of the commonly used uncertainty sets of robust optimization. We then reformulate the weighted-sum optimization problems of the multi-objective problems, derived with the aid of the optimality conditions, as equivalent conic linear programming problems, such as semidefinite programs or second-order cone programs, to find the (weak) Pareto optimal solutions. We illustrate by an example how our results can be used to find a second-stage (weak) Pareto optimal solution by solving a semidefinite program using a commonly available software.