Sensitivity Analysis for Piecewise-Affine Approximations of Nonlinear Programs with Polytopic Constraints

TL;DR

This paper develops a method to bound the solution differences between original nonlinear programs with polytopic constraints and their piecewise-affine approximations, aiding in efficient control system design.

Contribution

It introduces a perturbation analysis framework using convex modulus to derive guaranteed solution bounds for PWA approximations of NLPs with polytopic constraints.

Findings

Derived bounds on solution differences between NLP and PWA approximations.

Validated theoretical bounds through case studies on the Eggholder function and inverted pendulum control.

Abstract

Nonlinear Programs (NLPs) are prevalent in optimization-based control of nonlinear systems. Solving general NLPs is computationally expensive, necessitating the development of fast hardware or tractable suboptimal approximations. This paper investigates the sensitivity of the solutions of NLPs with polytopic constraints when the nonlinear continuous objective function is approximated by a PieceWise-Affine (PWA) counterpart. By leveraging perturbation analysis using a convex modulus, we derive guaranteed bounds on the distance between the optimal solution of the original polytopically-constrained NLP and that of its approximated formulation. Our approach aids in determining criteria for achieving desired solution bounds. Two case studies on the Eggholder function and nonlinear model predictive control of an inverted pendulum demonstrate the theoretical results.