A linear-time algorithm to compute the conjugate of nonconvex bivariate piecewise linear-quadratic functions

TL;DR

This paper introduces the first linear-time algorithm for computing the conjugate of nonconvex bivariate piecewise linear-quadratic functions, enabling efficient analysis of such functions in optimization problems.

Contribution

The paper presents a novel linear-time algorithm for conjugate computation of nonconvex bivariate PLQ functions, including implementation details and symbolic computation techniques.

Findings

Algorithm runs in linear time on triangulated subdivisions.

Open-source MATLAB implementation with symbolic computation.

Efficiently computes conjugates as piecewise quadratic functions.

Abstract

We propose the first linear-time algorithm to compute the conjugate of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm starts with computing the convex envelope of each quadratic piece obtaining rational functions (quadratic over linear) defined over a polyhedral subdivision. Then we compute the conjugate of each resulting piece to obtain piecewise quadratic functions defined over a parabolic subdivision. Finally we compute the maximum of all those functions to obtain the conjugate as a piecewise quadratic function defined on a parabolic subdivision. The resulting algorithm runs in linear time if the initial subdivision is a triangulation (or has a uniform upper bound on the number of vertexes for each piece). Our open-source implementation in MATLAB uses symbolic computation and rational numbers to avoid floating-point errors, and merges pieces as soon as possible to minimize computation time.