New complementarity formulations for root-finding and optimization of piecewise-affine functions in abs-normal form

TL;DR

This paper introduces new complementarity formulations for root-finding and optimization of piecewise-affine functions in abs-normal form, reducing restrictions and enabling efficient solution verification, with practical implementation in Julia.

Contribution

It presents novel, less restrictive complementarity formulations for piecewise-affine functions in abs-normal form, including methods to verify solution existence and a Julia implementation.

Findings

Piecewise-affine root-finding can be represented as a mixed-linear complementarity problem.

The approach often simplifies to a linear complementarity problem.

A Julia implementation demonstrates practical applicability.

Abstract

Nonsmooth functions have been used to model discrete-continuous phenomena such as contact mechanics, and are also prevalent in neural network formulations via activation functions such as ReLU. At previous AD conferences, Griewank et al. showed that nonsmooth functions may be approximated well by piecewise-affine functions constructed using an AD-like procedure. Moreover, such a piecewise-affine function may always be represented in an "abs-normal form", encoding it as a collection of four matrices and two vectors. We present new general complementarity formulations for root-finding and optimization of piecewise-affine functions in abs-normal form, with significantly fewer restrictions than previous approaches. In particular, piecewise-affine root-finding may always be represented as a mixed-linear complementarity problem (MLCP), which may often be simplified to a linear complementarity problem (LCP). We also present approaches for verifying existence of solutions to these problems. A proof-of-concept implementation in Julia is discussed and applied to several numerical examples, using the PATH solver to solve complementarity problems.