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.

我们研究了为最大化凸函数而定义的鲁棒优化问题。虽然问题出现在自然建模为最小化凹函数的情况下，但当决策者采取乐观的方法用凸函数做出决策时也会出现问题。对于有限不确定集，我们开发了一种几何分支定界算法方法来解决这个问题。几何分支定界算法执行凸目标的连续分段线性逼近，并求解线性程序以确定每个节点的下限和上限。算法的有限收敛到\(\epsilon -\)证明了最优解。数值结果用于讨论所开发算法的性能。本文所开发的算法可用作切割面法中求解具有紧凑模糊集的鲁棒优化问题的预言机。