Duality for non-convex variational principles
type: Item
instance of: scholarly article
title: Duality for non-convex variational principles
zbMATH Open document ID: 0533.49007
DOI: 10.1016/0022-0396(83)90085-2
published in: Journal of Differential Equations
publication date: 1983-01-01T00:00:00Z
review text: The author discusses different ways of constructing a dual problem to the problem of minimizing a given function f(x) on a locally convex space X. Any function \(L(x,y^*)\) such that \(f(x)=\sup_{y^*\in Y^*}L(x,y^*)\) is called a Lagrangian of type I for the problem \(\min_{x\in X}f(x)\). Together with the classical dual problem in which the function \(g(y^*)=\inf_{x}L(x,y^*)\) is maximized on the conjugate space \(Y^*\), the author considers an ''anomalous'' dual problem in which one minimizes on \(Y^*\) the function \(h(y^*)=\sup_{x}L(x,y^*)\). This makes sense because, as shown by the author, sometimes the two numbers \(\inf_{x}\sup_{y^*} L(x,y^*)\) and \(\inf_{y^*}\sup_{x}L(x,y^*)\) are equal. The function \(L(x,y^*)\) is said to be a Lagrangian of type II for the problem \(\min_{x}f(x)\) if \(f(x)=\inf_{y^*} L(x,y^*)\). In case such a Lagrangian is given, a dual problem can be constructed which consists in minimizing the function \(g(y^*)\) on \(Y^*\). All these duality problems are illustrated in various situations including the case when \(f(x)=F(x,Ax),\) where \(F:X\times Y\to R\) is a given real-valued function and A:\(X\to Y\) is a given continuous linear operator. Much attention is given to the critical points of Lagrangians and to Hamilton's equations. The last five sections (our of 12) are devoted to particular problems (and their duals) connected with: Rayleigh principle, convex programming, semilinear operator equations, classical dynamical systems and nonlinear elliptic problems.
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 49R50
Mathematics Subject Classification ID: 49S05
zbMATH DE Number: 3846317
zbMATH Keywords: Lagrangian duality
zbMATH Keywords: Hamilton's equations
zbMATH Keywords: Rayleigh principle
zbMATH Keywords: semilinear operator equations
author: Giles Auchmuty
reviewed by: Petar S. Kenderov
MaRDI profile type: Publication
full work available at URL: 0022-0396(83)90085-2
OpenAlex ID: W2007717983
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description: scientific article; zbMATH DE number 3846317