2017/06/01 by Tiến-Sơn Phạm, Pham, Tien-Son · 1 citation
Computer Science · Mathematics · #90C26 #90C30 #90C46 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1706.00234
openalex publication_date 2017/06/01 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this paper we study necessary optimality conditions for the optimization\nproblem
textrminfimumf0(x)
quad
textrm subject to
quad x
in S,\nwhere f0 colon \ℝn \→ \ℝ is a polynomial function\nand S \⊂ \ℝn is a set defined by polynomial inequalities.\nAssume that the problem is bounded below and has the Mangasarian--Fromovitz\nproperty at infinity. We first show that if the problem does em not have an\noptimal solution, then a version at infinity of the Fritz-John optimality\nconditions holds. From this we derive a version at infinity of the\nKarush--Kuhn--Tucker optimality conditions. As applications, we obtain a\nFrank--Wolfe type theorem which states that the optimal solution set of the\nproblem is nonempty provided the objective function f0 is convenient.\nFinally, in the unconstrained case, we show that the optimal value of the\nproblem is the smallest critical value of some polynomial. All the results are\npresented in terms of the Newton polyhedra of the polynomials defining the\nproblem.\n