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Optimality conditions in mathematical programming and composite optimization

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Abstract

New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat's rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki (1988) and Cominetti (1989); but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. We point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. Our conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral we recover the classical conditions of Ben-Tal—Zowe (1982) and Cominetti (1990).

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  1. Laboratoire de Mathématiques Appliquées (I.P.R.A.), University of Pau — URA 1204 CNRS, Av. de l'Université, 64000, Pau, France

    Jean-Paul Penot

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Penot, JP. Optimality conditions in mathematical programming and composite optimization. Mathematical Programming 67, 225–245 (1994). https://doi.org/10.1007/BF01582222

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  • DOI: https://doi.org/10.1007/BF01582222

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