Robust Fractional Programming

We extend robust optimization (RO) to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use RO. Our contribution is threefold. First, we provide conditions to guarantee that either a globally optimal solution, or a sequence converging to the globally optimal solution, can be found by solving one or more convex optimization problems. Second, we identify two cases for which an exact solution can be obtained by solving a single optimization problem: (1) when uncertainty in the numerator is independent from the uncertainty in the denominator, and (2) when the denominator does not contain an optimization variable. Third, we show that the general problem can be solved with an (iterative) root finding method. The results are demonstrated on a return on investment maximization problem, data envelopment analysis, and mean-variance optimization. We find that the robust optimal solution is only slightly more robust than the nominal solution. As a side-result, we use RO to show that two existing methods for solving fractional programs are dual to each other.