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

Leila Gharavi, Changrui Liu, Bart De Schutter, Simone Baldi

TL;DR

The paper addresses bounding the distance between the optimal solution of a polytopically constrained NLP and that of a continuous PWA approximation of its objective function. It leverages MMPS representations to obtain a locally convex approximation and uses the convexity modulus h1 to derive guaranteed bounds on minimizer deviations, quantified via the confidence radius χ. The main theoretical result provides an explicit bound ||hat{x}_p^* − x_p^*|| ≤ (2 Δ_p)/c1 + diam(C_{p,.}) and is validated through an Eggholder function case and NMPC for an inverted pendulum, illustrating how approximation quality and subregion geometry influence solution guarantees. The framework offers design criteria for PWA construction to meet targeted solution accuracy and supports offline tuning for tractable, reliable optimization in control applications.

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.

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

TL;DR

The paper addresses bounding the distance between the optimal solution of a polytopically constrained NLP and that of a continuous PWA approximation of its objective function. It leverages MMPS representations to obtain a locally convex approximation and uses the convexity modulus h1 to derive guaranteed bounds on minimizer deviations, quantified via the confidence radius χ. The main theoretical result provides an explicit bound ||hat{x}_p^* − x_p^*|| ≤ (2 Δ_p)/c1 + diam(C_{p,.}) and is validated through an Eggholder function case and NMPC for an inverted pendulum, illustrating how approximation quality and subregion geometry influence solution guarantees. The framework offers design criteria for PWA construction to meet targeted solution accuracy and supports offline tuning for tractable, reliable optimization in control applications.

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.
Paper Structure (12 sections, 8 theorems, 46 equations, 4 figures)

This paper contains 12 sections, 8 theorems, 46 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation of Continuous PWA Functions
  4. Sensitivity of the Convex Optimization Problem
  5. Continuous PWA Approximation of NLPs
  6. Theoretical Analysis
  7. Case Study
  8. Case 1: Finding the Confidence Radii
  9. Case 2: Finding the Approximation Criteria
  10. Application for NMPC Optimization
  11. Discussion
  12. Conclusions

Key Result

Theorem 1

For a scalar-valued continuous PWA function $f$ as in Definition def:cpwa, there exist non-empty index sets $\mathcal{I}_P$ and $\mathcal{I}_{Q_p}$ such that for real numbers $b_{p,q}$ and vectors $a_{p,q} \in \mathbb{R}^n$.

Figures (4)

  • Figure 1: A conceptual example of approximating a nonlinear function $F$ with a continuous PWA approximation $f$ using the MMPS form in (\ref{['eq:fmmps']}).
  • Figure 2: Plots of the nonlinear objective function $F$ and its PWA approximation $f$.
  • Figure 3: Comparison of two different PWA approximations of the nonlinear function on $\mathcal{C}_{3,.}$.
  • Figure 4: Comparison of two PWA approximations of NMPC.

Theorems & Definitions (17)

  • Definition 1: Continuous PWA function Chua1988
  • Theorem 1: MMPS representation Deschutter2004
  • Definition 2: Convexity modulus Phu2010
  • Theorem 2: Theorem 4.5 in Phu2010
  • Definition 3: Confidence radius
  • Proposition 1: Proposition 2.2, 2.5 in Phu2010
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 7 more