Quantitative Sensitivity Bounds for Nonlinear Programming and Time-varying Optimization

TL;DR

This paper develops explicit sensitivity bounds for solution maps of nonlinear and convex optimization problems, aiding the analysis of time-varying optimization in applications like power systems and robotics.

Contribution

It introduces new quantitative bounds for solution sensitivities, explicit Lipschitz constants, and a novel continuous-time algorithm modeled as a perturbed sweeping process.

Findings

Derived explicit local and global Lipschitz constants.

Bounded the rate of change of optimizers in time-varying problems.

Established asymptotic tracking bounds for a new discontinuous algorithm.

Abstract

Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit expressions for the local and global Lipschitz constants of the solution map of non-convex or convex optimization problems, respectively. Our results are geared towards the study of time-varying optimization problems which are commonplace in various applications of online optimization, including power systems, robotics, signal processing and more. In this context, our results can be used to bound the rate of change of the optimizer. To illustrate the use of our sensitivity bounds we generalize existing arguments to quantify the tracking performance of continuous-time, monotone running algorithms. Further, we introduce a new continuous-time running algorithm for time-varying constrained optimization which we model as a so-called perturbed sweeping process. For this discontinuous scheme, we establish an explicit bound on the asymptotic solution tracking for a class of convex problems.