Convex envelopes of bounded monomials on two-variable cones

• 成果类型: Article
• 署名作者: Belotti, Pietro
• 署名单位: Polytechnic University of Milan
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02212-5
• 发表日期: 2025
• 页码: 93-123
• 关键词: design; optimization

摘要

We consider an n-variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Monomial functions are building blocks for the class of Mixed Integer Nonlinear Optimization problems, which has many practical applications. We show that the upper envelope of the function in the given domain, for n >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{n}\ge 2$$\end{document}, is given by a conic inequality, and present the lower envelope for n=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {n=2}$$\end{document}. We also discuss branching rules that maintain these convex envelopes and their applicability in a branch-and-bound framework, then derive the volume of the convex hull for n=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {n=2}$$\end{document}.