Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions

Henri Bonnel; Ngoc Sang Pham

doi:10.3934/jimo.2011.7.789

Article Contents

2011, Volume 7, Issue 4: 789-809. Doi: 10.3934/jimo.2011.7.789

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Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions

Henri Bonnel^1, and
Ngoc Sang Pham^1,

1.
Equipe de Recherhce en Informatique et Mathématiques (ERIM), University of New Caledonia (France), B.P. R4, F98851, Nouméa Cedex

Received: September 30, 2010

Revised: April 30, 2011

Published: September 2011

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Abstract

We present explicit optimality conditions for a nonsmooth functional defined over the (properly or weakly) Pareto set associated with a multi-objective linear-quadratic control problem. This problem is very difficult even in a finite dimensional setting , i.e. when, instead of a control problem, we deal with a mathematical programming problem. Amongst various applications, our problem may be considered as a response for a decision maker when he has to choose a solution over the solution set of the grand coalition $p$-player cooperative differential game.

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Mathematics Subject Classification: Primary: 90C29, 91A23, 49K30; Secondary: 49N10, 91A12.

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