Solving cone-constrained convex programs by differential inclusions
type: Item
instance of: scholarly article
title: Solving cone-constrained convex programs by differential inclusions
zbMATH Open document ID: 0823.90098
DOI: 10.1007/BF01581692
author: Sjur Didrik Flåm
author: Alberto Seeger
published in: Mathematical Programming. Series A. Series B
publication date: 1994-09-26T00:00:00Z
review text: Let (P) be the problem of minimizing a continuous convex function \(f_ 0: X\to R\) subject to the constraint \(- F(x)\in K\), where \(X\) is a subset of a Hilbert space and \(K\) is a closed acute convex cone strictly contained in another Hilbert space. An exterior vector-valued penalization for the triple \((F, K, X)\) is a function \(f\) between the two spaces such that \(f(X)\subset K\) and \(f(x)= 0\) if and only if \(- F(x)\in K\); one assumes that the function \(\langle y, f(\cdot)\rangle\) is convex for all \(y\in K^*\), \(K^*\) being the positive polar cone of \(K\) and \(\langle\cdot,\cdot\rangle\) denoting inner product. The Lagrangian \(L\) associated to the penalization is given by \(L(x, y)= f_ 0(x)+ \langle y, f(x)\rangle\). To solve (P), the authors consider the differential inclusion \[ \dot x(t)\in -\partial_ x L(x(t), y(t))- N_ x(x(t)),\qquad \dot y(t)\in \partial_ y L(x(t), y(t))\tag{DI} \] together with the viability condition \(y(t)\in K^*\), where \(\partial_ x\) and \(\partial_ y\) are the partial subgradient operators and \(N_ X(x(t))\) stands for the normal cone to \(X\) at \(x(t)\). The existence and uniqueness of the trajectories of (DI) emanating from each point is established and their convergence to the set of saddle points of \(L\) is proved under suitable assumptions. An application to stochastic programming is given.
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 90C15
Mathematics Subject Classification ID: 90C48
zbMATH DE Number: 644746
zbMATH Keywords: saddle point
zbMATH Keywords: Hilbert space
zbMATH Keywords: differential inclusion
zbMATH Keywords: viability
zbMATH Keywords: partial subgradient operators
zbMATH Keywords: normal cone
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01581692
OpenAlex ID: W2079082939
description: scientific article; zbMATH DE number 644746