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Bounded Lagrange multiplier rules for general nonsmooth problems and application to mathematical programs with equilibrium constraints

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Abstract

In this paper, a general optimization problem is considered to investigate the conditions which ensure the existence of Lagrangian vectors with a norm not greater than a fixed positive number. In addition, the nonemptiness and boundedness of the multiplier sets together with their exact upper bounds is characterized. Moreover, three new constraint qualifications are suggested that each of them follows a degree of boundedness for multiplier vectors. Several examples at the end of the paper indicate that the upper bound for Lagrangian vectors is easily computable using each of our constraint qualifications. One innovation is introducing the so-called bounded Lagrangian constraint qualification which is stated based on the nonemptiness and boundedness of all possible Lagrangian sets. An application of the results for a mathematical program with equilibrium constraints is presented.

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, P. O. Box 81745-163, Isfahan, Iran

    Nooshin Movahedian

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  1. Nooshin Movahedian
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Correspondence to Nooshin Movahedian.

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The author would like to thank the Banach Algebra Center of Excellence for Mathematics, University of Isfahan.

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Movahedian, N. Bounded Lagrange multiplier rules for general nonsmooth problems and application to mathematical programs with equilibrium constraints. J Glob Optim 67, 829–850 (2017). https://doi.org/10.1007/s10898-016-0442-4

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  • DOI: https://doi.org/10.1007/s10898-016-0442-4

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