Techniques for exploring the suboptimal set

Abstract

The ε-suboptimal set $\mathcal{X}_{\epsilon}$ for an optimization problem is the set of feasible points with objective value within ε of optimal. In this paper we describe some basic techniques for quantitatively characterizing $\mathcal{X}_{\epsilon}$ , for a given value of ε, when the original problem is convex, by solving a modest number of related convex optimization problems. We give methods for computing the bounding box of $\mathcal{X}_{\epsilon}$ , estimating its diameter, and forming ellipsoidal approximations.

Quantitative knowledge of $\mathcal{X}_{\epsilon}$ can be very useful in applications. In a design problem, where the objective function is some cost, large $\mathcal{X}_{\epsilon}$ is good: It means that there are many designs with nearly minimum cost, and we can use this design freedom to improve a secondary objective. In an estimation problem, where the objective function is some measure of plausibility, large $\mathcal{X}_{\epsilon}$ is bad: It means that quite different parameter values are almost as plausible as the most plausible parameter value.