Gamma counterparts for robust nonlinear combinatorial and discrete optimization

TL;DR

This paper extends gamma uncertainty sets to robust nonlinear optimization, providing formulations that balance conservatism and tractability for mixed-integer nonlinear problems with various uncertainty structures.

Contribution

It generalizes gamma uncertainty sets to nonlinear and mixed-integer programs, deriving equivalent, tractable formulations for different uncertainty types and problem structures.

Findings

Robust counterparts are solvable with polynomial oracle calls in certain cases.

Formulations for linear, quadratic, and piecewise linear uncertainties are provided.

The approach balances robustness and computational efficiency.

Abstract

Gamma uncertainty sets have been introduced for adjusting the degree of conservatism of robust counterparts of (discrete) linear programs. The contribution of this paper is a generalization of this approach to (mixed integer) nonlinear optimization programs. We focus on the cases in which the uncertainty is linear or concave but also derive formulations for the general case. By applying reformulation techniques that have been established for nonlinear inequalities under uncertainty, we derive equivalent formulations of the robust counterpart that are not subject to uncertainty. The computational tractability depends on the structure of the functions under uncertainty and the geometry of its uncertainty set. We present cases where the robust counterpart of a nonlinear combinatorial program is solvable with a polynomial number of oracle calls for the underlying nominal program. Furthermore, we present robust counterparts for practical examples, namely for (discrete) linear, quadratic and piecewise linear settings. Keywords: Budget Uncertainty, Discrete Optimization, Combinatorial Optimization,