Robust Fractional Programming

Abstract

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.

Appendices

Appendix 1: The Importance of Convexity Conditions

We provide a short example to stress the importance of the convexity/concavity conditions on $f$ and $g$. The second numerical example in [8] is:

$$\begin{aligned} \min _{\mathbf {x} \in \mathbb {R}^n} \max _{a \in [0,1]} \quad \frac{ a^2 x_1 x_2 + x_1^{2a} + a x_3^3 }{ 5(a-1)^2 x_1^4 + 2x_2^2 + 4ax_3} \quad \text {s.t.} \quad 0.5 \le x_i \le 5, \quad i=1,2,3. \end{aligned}$$

This problem does not satisfy the convexity/concavity conditions from Sect. 4. Lin and Sheu claim that $\mathbf {x} = (0.5 \quad 1.5 \quad 0.5)$ and $a=0$ is optimal with a value of 0.21 (reported as $-0.21$), but $\mathbf {x} = (0.5 \quad 5 \quad 0.5)$ and $a=1$ is a better solution (maybe still not optimal) since the corresponding value is 0.06.

Appendix 2: On the Result by Kaul et al. [20]

This appendix shows a mistake in the paper by Kaul et al. [20]. Essentially, they formulate the dual of:

$$\begin{aligned} \min _{\alpha \in \mathbb {R}_+,\mathbf {x} \in \mathbb {R}^n_+} \; \alpha \quad \text {s.t.} \quad \frac{b_0 + \mathbf {b} ^{^{\top }}\mathbf {x}}{c_0 + \mathbf {c} ^{^{\top }}\mathbf {x}} \le \alpha , \; \forall (b_0,\mathbf {b}) \in \mathcal {U}_{1} \times \mathcal {U}_{2}, \; \forall (c_0,\mathbf {c}) \in \mathcal {U}_{3} \times \mathcal {U}_{4}, \\ \quad \mathbf {A} \mathbf {x} \le \mathbf {d}. \end{aligned}$$

Note that $\mathbf {x}$ is non-negative. In their Lemma 2.1, they claim that the worst case $(c_0,\mathbf {c})$ does not depend on $\mathbf {x}$, and is given by $c_0^* = \min _{c_0 \in \mathcal {U}_{3}} \{ c_0 \}$ and $\mathbf {c^*}$, with components $c_i^* = \min _{\mathbf {c} \in \mathcal {U}_{4}} \{ c_i \}$. This implicitly assumes that $\mathbf {c^*}$ is a member of $\mathcal {U}_{4}$, which is not always true. The mistake becomes clear in their numerical example, where they use $\mathbf {c^*} = [4; 2]$, which is not in the uncertainty set. Consequently, the proposed approach gives the wrong dual problem and a suboptimal solution. Our results in Sect. 4.2 can provide the correct dual problem under milder conditions on the uncertainty sets.