In this talk, we address robust optimization problems involving robust nonlinear constraints, that are defined as sums of products of linear times concave functions with respect to the uncertain parameters. We introduce the Reformulation-Perspectification Technique, which allows us to obtain a convex relaxation of such problems and subsequently solve them to optimality. We provide multiple examples of robust optimization problems falling in that class. Further, we extend the applicability of our approach to two-stage adaptive robust optimization problems.

In this talk, we study robust nonlinear optimization problems involving sum of linear times convex (SLC) functions as well as uncertainty sets that are defined by conic constraints belonging to one of the five basic cones, that is, linear cone, second order cone, power cone, exponential cone, and semidefinite cone. By using the Reformulation Perspectification Technique, we can obtain a convex relaxation by forming the perspective of each function and linearizing all product terms of uncertain parameters with newly introduced uncertain parameters. To further tighten the approximation, we can pairwise multiply the conic constraints. In this paper, we analyze all possibilities of multiplying conic constraints. Moreover, in case of an exponential cone we generate valid inequalities that can be used to further strengthen the approximation and in case of a power cone we generate additional valid inequalities.

We introduce a novel exact approach for solving optimization problems with robust nonlinear constraints, that are sums of products of linear times concave (SLC) functions with respect to the uncertain parameters. Our approach synergizes a cutting set method with reformulation-perspectification techniques and branch and bound. Numerical experiments on a robust convex geometric optimization problem and a robust linear optimization problem with data uncertainty and implementation error show that our approach can solve robust nonlinear problems that cannot be solved by existing methods in the literature. Moreover, a numerical experiment on a lot-sizing problem on a network demonstrates the efficiency of our method for two-stage ARO problems.