计算二元分段线性二次函数的凸包络的线性时间算法

中文摘要 AI 辅助

我们计算（非凸）二元分段线性二次（PLQ）函数（定义在多面体细分上的二元二次函数）的凸包络。我们的算法由以下步骤组成：（1）计算每个二次片段的凸包络，得到定义在多面体细分上的分段有理函数（二次函数除以线性函数）；（2）计算每个结果片段（勒让德-芬切尔）共轭，得到定义在抛物细分上的分段二次函数；（3）计算所有这些函数的最大值，得到原始PLQ函数的共轭，作为定义在抛物细分上的分段二次函数；（4）计算每个结果片段的共轭；最后（5）计算所有这些函数的最大值，得到凸包络（双共轭），作为定义在多面体细分上的有理函数（二次函数除以线性函数）。我们的贡献包括一个线性时间运行的实用算法，并证明凸包络是分段有理函数。

英文摘要

We compute the convex envelope of (nonconvex) bivariate piecewise linear-quadratic (PLQ) functions (bivariate quadratic functions defined on a polyhedral subdivision). Our algorithm is composed of the following steps: (1) compute the convex envelope of each quadratic piece obtaining piecewise rational functions (quadratic divided by linear function) defined over a polyhedral subdivision; (2) compute the (Legendre-Fenchel) conjugate of each resulting piece to obtain piecewise quadratic functions defined over a parabolic subdivision; (3) compute the maximum of all those functions to obtain the conjugate of the original PLQ function as a piecewise quadratic function defined on a parabolic subdivision; (4) compute the conjugate of each resulting piece; and finally (5) compute the maximum over all those functions to obtain the convex envelope (biconjugate) as rational functions (quadratic divided by linear function) defined over a polyhedral subdivision. Our contribution includes a practical algorithm running in linear time, and proving that the convex envelope is a piecewise rational function.

发表机构

• Computer Science, CMPS, I. K. Barber Faculty of Science, UBC Okanagan（不列颠哥伦比亚大学奥肯那根分校 I.K.巴伯理学院）

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