Towards the Biconjugate of Bivariate Piecewise Quadratic Functions

TL;DR

This paper presents a linear-time algorithm for computing the conjugate of bivariate piecewise quadratic functions over polytopes, advancing the understanding of convex envelopes and rational function conjugates.

Contribution

It introduces a novel method for computing the conjugate of bivariate piecewise quadratic functions with explicit rational and fractional forms.

Findings

Conjugate functions have a parabolic subdivision with fractional form.

The algorithm operates with worst-case linear time complexity.

Results facilitate explicit formulas for convex envelopes of piecewise rational functions.

Abstract

Computing the closed convex envelope or biconjugate is the core operation that bridges the domain of nonconvex with convex analysis. We focus here on computing the conjugate of a bivariate piecewise quadratic function defined over a polytope. First, we compute the convex envelope of each piece, which is characterized by a polyhedral subdivision such that over each member of the subdivision, it has a rational form (square of a linear function over a linear function). Then we compute the conjugate of all such rational functions. It is observed that the conjugate has a parabolic subdivision such that over each member of its subdivision, it has a fractional form (linear function over square root of a linear function). This computation of the conjugate is performed with a worst-case linear time complexity algorithm. Our results are an important step toward computing the conjugate of a piecewise quadratic function, and further in obtaining explicit formulas for the convex envelope of piecewise rational functions.