Optimality Conditions for Minimizers at Infinity in Polynomial Programming

• 成果类型: Article
• 署名作者: Tien-Son Pham
• 署名单位: Ton Duc Thang University; Ton Duc Thang University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2018.0974
• 发表日期: 2019
• 页码: 1381-1395
• 关键词: wolfe type theorem; convex; set

摘要

In this paper we study necessary optimality conditions for the problem of minimizing a polynomial function over a set defined by polynomial inequalities. Assume that the problem is bounded below and has the Mangasarian-Fromovitz property at infinity. We first show that if the problem does not have an optimal solution, then a version at infinity of the Fritz John optimality conditions holds. From this we derive a version at infinity of the Karush-Kuhn-Tucker optimality conditions. As applications, we obtain a Frank-Wolfe type theorem which states that the optimal solution set of the problem is nonempty provided the objective function is convenient. Finally, in the unconstrained case, we show that the optimal value of the problem is the smallest critical value of some polynomial. All the results are presented in terms of the Newton polyhedra of the polynomials defining the problem.