A Geometric Branch and Bound Method for a Class of Robust Maximization Problems of Convex Functions

TL;DR

This paper introduces a geometric branch-and-bound algorithm for robust maximization of convex functions with finite uncertainty sets, providing convergence guarantees and demonstrating practical performance.

Contribution

It develops a novel geometric branch-and-bound method that efficiently solves robust convex maximization problems with finite uncertainty sets, including convergence proof.

Findings

Algorithm converges to an $ ext{epsilon}$-optimal solution.

Numerical results show effective performance of the method.

The approach can serve as an oracle in cutting surface methods.

Abstract

We investigate robust optimization problems defined for maximizing convex functions. For finite uncertainty set, we develop a geometric branch-and-bound algorithmic approach to solve this problem. The geometric branch-and-bound algorithm performs sequential piecewise-linear approximations of the convex objective, and solves linear programs to determine lower and upper bounds of nodes specified by the active linear pieces. Finite convergence of the algorithm to an $\epsilon-$optimal solution is proved. Numerical results are used to discuss the performance of the developed algorithm. The algorithm developed in this paper can be used as an oracle in the cutting surface method for solving robust optimization problems with compact ambiguity sets.