Robust fractional programming

TL;DR

This paper extends Robust Optimization to fractional programming with uncertain parameters in both objectives and constraints, providing conditions for globally optimal solutions and practical solution methods, demonstrated through financial and analytical examples.

Contribution

It introduces new conditions and methods for solving robust fractional programming problems, including cases with exact solutions and iterative root-finding approaches, advancing the field's capabilities.

Findings

Robust solutions are only slightly more conservative than nominal solutions.

Two existing fractional programming methods are shown to be dual to each other.

The proposed methods are demonstrated on financial and analytical problems.

Abstract

We extend Robust Optimization 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 Robust Optimization. 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 Robust Optimization to show that two existing methods for solving fractional programs are dual to each other.