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

Tanmaya Karmarkar, Yves Lucet

TL;DR

This work addresses computing the conjugate of nonconvex bivariate PLQ functions defined on polyhedral domains. It introduces a three-step framework that first convexifies each piece, then conjugates the convex pieces, and finally takes a maximum to assemble the global conjugate on a parabolic subdivision; the algorithm achieves linear time under triangulated domains. Theoretical contributions include establishing that the conjugate is a piecewise quadratic function on a parabolic subdivision and demonstrating that fractional forms do not arise, along with a complete recipe for Step 3 to obtain the final conjugate. An open-source MATLAB implementation using symbolic rational arithmetic is provided to avoid floating-point errors and to merge adjacent pieces efficiently, with experiments validating correctness and illustrating performance trends.

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.

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

TL;DR

This work addresses computing the conjugate of nonconvex bivariate PLQ functions defined on polyhedral domains. It introduces a three-step framework that first convexifies each piece, then conjugates the convex pieces, and finally takes a maximum to assemble the global conjugate on a parabolic subdivision; the algorithm achieves linear time under triangulated domains. Theoretical contributions include establishing that the conjugate is a piecewise quadratic function on a parabolic subdivision and demonstrating that fractional forms do not arise, along with a complete recipe for Step 3 to obtain the final conjugate. An open-source MATLAB implementation using symbolic rational arithmetic is provided to avoid floating-point errors and to merge adjacent pieces efficiently, with experiments validating correctness and illustrating performance trends.

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.
Paper Structure (22 sections, 7 theorems, 30 equations, 16 figures, 7 tables)

This paper contains 22 sections, 7 theorems, 30 equations, 16 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computing Conjugates
  4. Algorithm
  5. Step 3a
  6. Step 3a(i)
  7. Step 3a(ii)
  8. Step 3a(iii)
  9. Division
  10. Step 3b
  11. Divisions
  12. Examples
  13. Verification
  14. Experiment 1
  15. Experiment 2
  16. ...and 7 more sections

Key Result

Proposition 1

Assume $f(x)=x^TAx+b^Tx+c +I_P$ with $P=\operatorname{conv}\{x_1,x_2,x_3\}$ a triangular set in $\mathbb{R}^2$. Then its convex envelope $\operatorname{conv} f = \min f_i + I_{P_i}$ is a piecewise function over a polyhedral subdivision where $f_i$ is quadratic, linear, or a rational function while $

Figures (16)

  • Figure 1: Illustration of the steps taken on the domains to compute the biconjugate of a PLQ function. Note that the biconjugate has a polyhedral subdivision, but an unknown explicit formula leading us to label the last box "Biconjugate ??".
  • Figure 2: Steps to compute the conjugate of a PLQ Function illustrated on the domain of the respective functions.(Functions defined in Table \ref{['table:allsteps']})
  • Figure 3: Intersection of domains of two conjugates.
  • Figure 4: Computing $\operatorname{dom} (s_{i,1,k})\cap\operatorname{dom} (s_{i,2,k})$ using $\operatorname{dom} (r_{i,1})\cap\operatorname{dom} (r_{i,2}).$
  • Figure 5: Further division by a parabolic curve as explained in Example \ref{['division']}.
  • ...and 11 more figures

Theorems & Definitions (23)

  • Example 1: Univariate PLQ function.
  • Example 2: Bivariate PLQ function.
  • Example 3: Convex Envelope of Example \ref{['uPLQ']}
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 4
  • Proposition 3
  • proof
  • ...and 13 more