boundedness – Optimization Online

On the boundedness of infinite products of relaxed projections: perturbations resilience and dynamic string-averaging

Published: 2026/07/29

Basic Sciences Applications, Convex Optimization, Infinite Dimensional Optimization Affine subspace, boundedness, control, dynamic string-averaging, half-space, infinite product, innately regular, perturbation resilience, relaxed projection, strongly nonexpansive mapping

Very recently (2026), Bauschke and Tung extended from finite- to infinite-dimensional Hilbert spaces a result published by Meshulam in 1996 (following an earlier result of Aharoni-Duchet-Wajnryb from 1984) regarding the boundedness of infinite products of relaxed projections onto a finite family of closed affine subspaces. In the present note we extend in various ways the … Read more

Weakly Homogeneous Optimization Problems

Published: 2020/01/28, Updated: 2020/02/17

Vu Trung Hieu

Nonlinear Optimization asymptotic cone, boundedness, nonemptiness, optimization problem, pseudoconvex function, upper semincontinuity, weakly homogeneous function

This paper investigates a new class of optimization problems whose objective functions are weakly homogeneous relative to the constrain sets. Two sufficient conditions for nonemptiness and boundedness of solution sets are established. We also study linear parametric problems and upper semincontinuity of the solution map. ArticleDownload View PDF

Complementary problems with polynomial data

Published: 2019/08/17

Complementarity and Variational Inequalities boundedness, error bound, existence, genericity, polynomial complementarity problem, uniqueness

Given polynomial maps $f, g \colon \mathbb{R}^n \to \mathbb{R}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x \in \mathbb{R}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad \textrm{ and } \quad \langle f(x), g(x) \rangle \ = \ 0. \end{equation*} In this paper, we … Read more

Tangencies and Polynomial Optimization

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

Tiên-So'n Pham

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