tangencies – 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

Local minimizers of semi-algebraic functions

Published: 2019/01/07

Tiên-So'n Pham

Nonsmooth Optimization, Unconstrained Optimization \l ojasiewicz gradient inequality, local minimizers, optimality conditions, semi-algebraic, sharp minimality, strong metric subregularity, tangencies

Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so–called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of … Read more