[Paper Review] Novel convex decomposition of piecewise affine functions

Nils Schlüter, Moritz Schulze Darup|arXiv (Cornell University)|Aug 9, 2021

Advanced Control Systems Optimization6 references4 citations

TL;DR

This paper proposes a novel optimization-based method to decompose any continuous piecewise affine (PWA) function into two convex PWA functions of equal complexity. Unlike prior approaches that suffer from high or imbalanced complexity, this method ensures applicability to all cases and significantly reduces computational overhead in explicit model predictive control (MPC) applications.

ABSTRACT

In this paper, we present a novel approach to decompose a given piecewise affine (PWA) function into two convex PWA functions. Convex decompositions are useful to speed up or distribute evaluations of PWA functions. Different approaches to construct a convex decomposition have already been published. However, either the two resulting convex functions have very high or very different complexities, which is often undesirable, or the decomposition procedure is inapplicable even for simple cases. Our novel methodology significantly reduces these drawbacks in order to extend the applicability of convex decompositions.

Motivation & Objective

• To address the limitations of existing convex decomposition methods for piecewise affine (PWA) functions, which often result in high or imbalanced complexity.
• To develop a method that is universally applicable, even for complex PWA functions where prior approaches fail or become computationally infeasible.
• To maintain the benefits of convex decomposition—such as faster evaluation and reduced memory usage—while ensuring both convex components have comparable complexity.
• To demonstrate the practical advantages in explicit model predictive control (MPC), where PWA functions are common and evaluation efficiency is critical.

Proposed method

• The method formulates the decomposition of a PWA function $ f(oldsymbol{x}) $ into $ g(oldsymbol{x}) - h(oldsymbol{x}) $, where both $ g $ and $ h $ are convex PWA functions.
• It uses an optimization-based approach to construct $ g $ and $ h $ such that their underlying partitions are balanced in size, avoiding the high complexity seen in convex folds or hyperplane arrangement methods.
• The approach ensures that the number of polyhedral segments in both $ g $ and $ h $ is minimized and equal, based on a structured refinement of the initial partition $ igcup_{i=1}^s ilde{oldsymbol{X}}_i $.
• The method is designed to be always applicable, even when previous approaches fail—such as in cases with overlapping or complex polyhedral regions.
• It leverages the fact that convex PWA functions can be evaluated efficiently via max-of-affine representations, enabling faster evaluation through parallel or distributed computation.
• The decomposition is validated using an explicit MPC benchmark with varying prediction horizons $ N $, where the complexity of resulting partitions is measured and compared.

Experimental results

Research questions

• RQ1 Can a convex decomposition of a PWA function be constructed such that both convex components have equal and minimal complexity?
• RQ2 Is it possible to design a decomposition method that is universally applicable, even in cases where existing methods fail or become numerically intractable?
• RQ3 How does the proposed method compare in terms of computational complexity and evaluation efficiency to existing approaches like convex folds and hyperplane arrangement?
• RQ4 To what extent can the proposed decomposition reduce evaluation time and memory usage in explicit MPC applications?

Key findings

• The proposed method achieves equal complexity for both convex components $ g $ and $ h $, with 339 polyhedral segments for $ N = 10 $, compared to 105,581 and 106,697 in the convex folds method.
• For $ N = 15 $, the total complexity of the novel decomposition (694 segments) is lower than that of the convex folds method (803 segments), demonstrating superior efficiency.
• The optimization-based method from [5] becomes impractical for $ N > 1 $, requiring hyperplane arrangement regularization that increases complexity to 26,786 segments for $ N = 15 $.
• Evaluation time was reduced by a factor of 10, and storage capacity was reduced by a factor of 16 in the explicit MPC case study, due to efficient max-of-affine evaluation of $ g $ and $ h $.
• The method is always applicable, even for complex PWA functions where prior approaches fail or produce highly unbalanced decompositions.
• The decomposition enables faster evaluation via max-of-affine computation, outperforming traditional binary search tree methods in explicit MPC.