existence of minimizers – Optimization Online

Tangencies and Polynomial Optimization

Published: 2019/02/16, Updated: 2019/03/09

Constrained Nonlinear Optimization, Global Optimization Theory boundedness, coercivity, compactness, critical points, existence of minimizers, polynomial, semi-algebraic, stability, sub-levels, tangencies

Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The … Read more

Optimality conditions for minimizers at infinity in polynomial programming

Published: 2017/06/01, Updated: 2017/06/29

Tiên-So'n Pham

Constrained Nonlinear Optimization, Global Optimization existence of minimizers, fermat theorem, frank-wolfe, fritz-john optimality conditions, karush--kuhn--tucker optimality conditions, mangasarian-fromovitz constraint qualification, newton polyhedron, polynomial programming

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 Mangasarian–Fromovitz property … Read more