Techniques for Exploring the Suboptimal Set

The ǫ-suboptimal set Xǫ for an optimization problem is the set of feasible points with objective value within ǫ of optimal. In this paper we describe some basic techniques for quantitatively characterizing Xǫ, for a given value of ǫ, 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 Xǫ, estimating its diameter, and forming ellipsoidal approximations. Quantitative knowledge of Xǫ can be very useful in applications. In a design problem, where the objective function is some cost, large Xǫ 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 Xǫ is bad: It means that quite different parameter values are almost as plausible as the most plausible parameter value. 1 The ǫ-suboptimal set We consider the general convex optimization problem minimize f(x) subject to x ∈ C (1) where x ∈ R is the optimization variable, f is the (convex) objective function, and C is the (closed convex) feasible set. We assume that p, the optimal value of the problem, is finite, and achieved by at least one point. For ǫ ≥ 0, we say that x is ǫ-suboptimal if x ∈ C and f(x) ≤ p + ǫ. We will let Xǫ denote the set of all ǫ-suboptimal points, Xǫ = {x ∈ C | f(x) ≤ p + ǫ}, (2) This material is based upon work supported by the Focus Center Research Program Center for Circuit & System Solutions award #2003-CT-888, by JPL award #I291856, by Army award #W911NF-07-1-0029, by NSF award #ECS-0423905, by NSF award #0529426, by DARPA award #N66001-06-C-2021, by NASA award #NNX07AEIIA, by AFOSR award #FA9550-06-1-0514, and by AFOSR award #FA9550-06-1-0312. The authors are with the Information Systems Lab, Electrical Engineering Department, Stanford University, Stanford, CA 94305-9510 USA. Email: {jskaf,boyd}@stanford.edu.