Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers
type: Item
instance of: scholarly article
title: Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers
zbMATH Open document ID: 1433.90158
published in: Journal of Global Optimization
publication date: 2018-12-27T00:00:00Z
review text: The authors provide a new generalized Lagrange duality concept for (not necessary convex) optimization problems where the objective function and the constraint functions are polynomials and where also the Lagrange multipliers are replaced by polynomials. In detail, let \(\mathbb{R}[x]\) be the ring of real polynomials in \(x\in\mathbb{R}^n\) and \(\sum_n\) be the set of functions which are the sum of square polynomials, i.e., functions \(f=\sum^r_{j=1}(f_j)^2\) with \(f_j\in\mathbb{R}[x]\). Starting with the primal optimization problem \[f(x)\to\min\quad\text{s.t.}\quad h_i(x)= 0\,\, (i= 1,\dots,p),\quad g_j(x)\ge 0\,\, (j=1,\dots,m)\] (where all functions \(f\), \(h_i\), \(g_j\in\mathbb{R}[x]\)), the associated polynomial Lagrangian function is introduced by \[L(x,\phi,\sigma):=f(x)- \sum^p_{i=1} \phi_i(x) h)i(x)- \sum^m_{j=1} \sigma_j(x) g_j(x)\] and the dual optimization problem by \[\inf\{L(x,\phi,\sigma\mid x\in\mathbb{R}^n\}\to\max\quad\text{s.t. }\phi_i\in \mathbb{R}[x]\ (i= 1,\dots,p),\ \sigma_j\in\sum_n\ (j= 1,\dots,m).\] Using a suitable nonnegativity certificate, the authors provide necessary and sufficient optimality conditions, but also weak and strong duality assertions in connection with the presence of saddle points of the introduced Lagrangian. Several numerical examples illustrate the results.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 90C29
zbMATH DE Number: 6996809
zbMATH Keywords: nonconvex polynomial programs
zbMATH Keywords: generalized Lagrangian duality
zbMATH Keywords: global optimality
zbMATH Keywords: sum of squares polynomials
zbMATH Keywords: quadratic programs
zbMATH Keywords: separable programs
author: Thai Doan Chuong
author: V. Jeyakumar
describes a project that uses: SDPT3
describes a project that uses: CVX
MaRDI profile type: Publication
full work available at URL: s10898-018-0665-7
OpenAlex ID: W2803892878
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DOI: 10.1007/S10898-018-0665-7
description: scientific article; zbMATH DE number 6996809