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A sharp Lagrange multiplier theorem for nonlinear programs

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Abstract

For a nonlinear program with inequalities and under a Slater constraint qualification, it is shown that the duality between optimal solutions and saddle points for the corresponding Lagrangian is equivalent to the infsup-convexity—a not very restrictive generalization of convexity which arises naturally in minimax theory—of a finite family of suitable functions. Even if we dispense with the Slater condition, it is proven that the infsup-convexity is nothing more than an equivalent reformulation of the Fritz John conditions for the nonlinear optimization problem under consideration.

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Acknowledgments

Research partially supported by Junta de Andalucía Grant FQM359 and FEDER.

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  1. Department of Applied Mathematics, E.T.S. Ingeniería de Edificación, University of Granada, c/ Severo Ochoa s/n, 18071, Granada, Spain

    M. Ruiz Galán

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  1. M. Ruiz Galán
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Correspondence to M. Ruiz Galán.

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Galán, M.R. A sharp Lagrange multiplier theorem for nonlinear programs. J Glob Optim 65, 513–530 (2016). https://doi.org/10.1007/s10898-015-0379-z

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