# Towards the Biconjugate of Bivariate Piecewise Quadratic Functions

• Deepak Kumar
• Yves Lucet
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

## 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.

## Keywords

Conjugate Convex envelope Piecewise quadratic function

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## Authors and Affiliations

• Deepak Kumar
• 1
• Yves Lucet
• 1
1. 1.University of British Columbia OkanaganKelownaCanada