An objective penalty function method for biconvex programming
type: Item
instance of: scholarly article
title: An objective penalty function method for biconvex programming
zbMATH Open document ID: 1487.90534
author: Zhiqing Meng
author: Min Jiang
author: R. Shen
author: Leiyan Xu
author: Chuangyin Dang
published in: Journal of Global Optimization
publication date: 2021-11-26T00:00:00Z
review text: A function \(f(x_1,x_2)\) is biconvex when \(f(\cdot,x_2)\) and \(f(x_1,\cdot)\) are both convex for all \(x_1,x_2\). A point \((x_1^*,x_2^*)\) is said to be a partial minimiser if \(x_1^*\) is a minimiser of \(f(\cdot, x_2^*)\) and \(x_2^*\) is a minimiser of \(f(x_1^*,\cdot)\). Consider the constrained biconvex programming problem of minimising the function \(f\) subject to the constraint \(g(x_1,x_2)\leq 0\) where \(g\) is also biconvex. One can similarly define the notion of partial KKT solutions as points satisfying KKT conditions in each of its variables, when the other one is fixed (both subproblems are then convex). A penalisation of this problem consisting of minimising \(P(f(x_1,x_2)) + \rho Q(g(x_1,x_2))\) is a partially exact objective penalty function if it admits a partial optimum that coincides with the partial optimum of the original constrained problem for some \(\rho\). This paper presents algorithms based on properties of biconvex function relating to their partial properties. The first part establishes the equivalence between partial optimality and the satisfaction of partial KKT conditions, under reasonable conditions on \(P\) and \(Q\). This result opens the possibility for ADMM-like algorithms that converge to partial minimisers to constrained biconvex problems using a penalty approach. This is the focus of the second part of the paper.
reviewed by: Julien Ugon
Mathematics Subject Classification ID: 90C26
zbMATH DE Number: 7433931
zbMATH Keywords: biconvex programming
zbMATH Keywords: objective penalty function
zbMATH Keywords: partial optimum
zbMATH Keywords: partial exactness
zbMATH Keywords: partial stability
describes a project that uses: Pyglrm
describes a project that uses: LowRankModels
MaRDI profile type: Publication
full work available at URL: s10898-021-01064-5
OpenAlex ID: W3197628960
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DOI: 10.1007/S10898-021-01064-5
description: scientific article; zbMATH DE number 7433931