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TienSon Pham (sonptdlu.edu.vn) Abstract: In this paper we study necessary optimality conditions for the optimization problem $$\textrm{infimum}f_0(x) \quad \textrm{ subject to } \quad x \in S,$$ where $f_0 \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is a polynomial function and $S \subset \mathbb{R}^n$ is a set defined by polynomial inequalities. Assume that the problem is bounded below and has the MangasarianFromovitz property at infinity. We first show that if the problem does {\em not} have an optimal solution, then a version at infinity of the FritzJohn optimality conditions holds. From this we derive a version at infinity of the KarushKuhnTucker optimality conditions. As applications, we obtain a FrankWolfe type theorem which states that the optimal solution set of the problem is nonempty provided the objective function $f_0$ 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. Keywords: Existence of minimizers, Fermat theorem, FrankWolfe theorem, FritzJohn optimality conditions, KarushKuhnTucker optimality conditions, MangasarianFromovitz constraint qualification, Newton polyhedron, polynomial programming Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization ) Category 2: Global Optimization Citation: Download: [PDF] Entry Submitted: 06/01/2017 Modify/Update this entry  
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