  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.
