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Inclusion conditions for the Constrained Polynomial Zonotopic case

Bogdan Gheorghe, Amr Alanwar, Florin Stoican

Abstract

Set operations are well understood for convex sets but become considerably more challenging in the non-convex case due to the loss of structural properties in their representation. Constrained polynomial zonotopes (CPZs) offer an effective compromise, as they can capture complex, typically non-convex geometries while maintaining an algebraic structure suitable for further manipulation. Building on this, we propose novel nonlinear encodings that provide sufficient conditions for testing inclusion between two CPZs and adapt them for seamless integration within optimization frameworks.

Inclusion conditions for the Constrained Polynomial Zonotopic case

Abstract

Set operations are well understood for convex sets but become considerably more challenging in the non-convex case due to the loss of structural properties in their representation. Constrained polynomial zonotopes (CPZs) offer an effective compromise, as they can capture complex, typically non-convex geometries while maintaining an algebraic structure suitable for further manipulation. Building on this, we propose novel nonlinear encodings that provide sufficient conditions for testing inclusion between two CPZs and adapt them for seamless integration within optimization frameworks.

Paper Structure

This paper contains 5 sections, 3 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on zonotopes
  3. Inclusion conditions for constrained polynomial zonotopes
  4. Simulation results for constrained polynomial zonotopes inclusion
  5. Conclusions

Key Result

Proposition 1

Consider the constrained polynomial zonotopic sets $\mathcal{P}_i = \langle c_i, G_i, E_i, F_i, \theta_i, R_i\rangle_{CPZ} \in (\mathbb R^{d}, \mathbb R^{d \times n_i}, \mathbb N_0^{s_i \times n_i}, \mathbb \mathbb{R}^{^}{p_i\times q_i}, \mathbb R^{p_i}, \mathbb N_0^{s_i \times q_i})$ for $i \in \{1 are sufficient to guarantee the inclusion $\mathcal{P}_1 \subseteq \mathcal{P}_2$.

Figures (2)

  • Figure 1: Different zonotopic representations
  • Figure 2: CPZ sets used for the inclusion validation.

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Remark 3
  • Corollary 2
  • proof
  • ...and 1 more