The Flexible Γ-Approach for Nonlinear Discrete and Nonlinear Combinatorial Optimization

The flexible Γ-approach has been introduced for adjusting the degree of conservatism in robust counterparts. It has mainly been applied to linear combinatorial optimization problems: Instead of aiming for solutions which are optimal regardless of how the uncertainties manifest, the objective is to ensure robustness against Γ uncertainties. The contribution of this paper is a generalization of this approach for (mixed-integer) nonlinear optimization problems. We study the cases in which the functions considered are concave or linear, as well as non-concave, in the uncertainty. By applying reformulation techniques that have been established for nonlinear inequalities under uncertainty, we derive equivalent robust counterparts. In both cases the computational tractability of the counterpart depends on the structure of the geometry of the uncertainty set. We explicitly present robust counterparts for combinatorial problems, e.g. the Quadratic Assignment Problem. We conduct computational studies for the Γ-robust Quadratic Assignment Problem and the Γ-robust Vehicle Routing Problem with Soft Time Windows to demonstrate the computational tractability in terms of solution quality and running time.