Over-Approximating Minimizer Sets of Constrained Convex Programs with Parametric Uncertainty via Reachability Analysis

TL;DR

This paper presents a method to compute certified outer approximations of the solution sets of convex optimization problems with uncertain parameters by modeling PGD as a dynamical system and analyzing its reachable sets.

Contribution

It introduces a reachability analysis approach using system-level synthesis to efficiently over-approximate minimizer sets under parametric uncertainty.

Findings

The method provides exponential decay of approximation error with iterations.

It outperforms existing baselines with lower conservativeness.

Applicable to high-dimensional decision variables in convex programs.

Abstract

We study the set of solutions to a parameterized, strongly convex optimization problem whose cost depends on uncertain, bounded parameters. We compute a certified outer approximation of the corresponding set of optimizers, using convergence properties of the projected gradient descent (PGD) algorithm for convex programs. Concretely, by treating the cost parameter as constant but unknown, we interpret the PGD iterates as an uncertain dynamical system and analyze its forward reachable sets. Since PGD converges exponentially to the unique optimizer for each fixed parameter, these reachable sets provide outer approximations of the optimizer set, with an explicit error bound that decays exponentially with the iteration count. We apply system-level synthesis (SLS) on the PGD dynamics to optimize the step-size sequence and obtain reachable-set over-approximations. Our method outperforms existing baselines in over-approximating, with low conservativeness, the minimizer sets of convex programs with uncertain costs and high-dimensional decision variables.