A geometric branch and bound method for robust maximization of convex functions

We investigate robust optimization problems defined for maximizing convex functions. While the problems arise in situations which are naturally modeled as minimization of concave functions, they also arise when a decision maker takes an optimistic approach to making decisions with 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 at each node. 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.