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Optimality conditions for nonlinear infinite programming problems

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Abstract

An infinite programming problem consists in minimizing a functional defined on a real Banach space under an infinite number of constraints. The main purpose of this article is to provide sufficient conditions of optimality under generalized convexity assumptions. Such conditions are necessarily satisfied when the problem possesses the property that every stationary point is a global minimizer.

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Notes

  1. See Folland [9].

  2. See, for example, Brézis [3] and Folland [9].

  3. We denote by \({\mathcal {B}}(A)\) the Borel \(\sigma \)-algebra on \(A\).

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Acknowledgments

The authors gratefully acknowledge the valuable comments of anonymous reviewers, which certainly improved the final presentation of this paper. V. A. de Oliveira was supported by grant 2011/01977-2, São Paulo Research Foundation (FAPESP). L. B. dos Santos was partially supported by the Spanish Ministry of Education and Science MEC Spain—Grant MTM2010-15383, and by CAPES Brasil—Grant BEX 9535-11-0. R. Osuna-Gómez was supported by Ministerio de Ciencia e Innovación through MTM2010-15383. M. A. Rojas-Medar was partially supported by the Spanish Ministry of Education and Science (MEC)—Grant MTM2010-15383, by the National Fund for Scientific and Technological Development (Fondecyt-Chile)—Project 1120260 and GI/C-UBB 121909.

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Authors and Affiliations

  1. Instituto de Biociências, Letras e Ciências Exatas, UNESP - Univ. Estadual Paulista, Rua Cristóvão Colombo, N. 2265, São José do Rio Preto, SP, CEP 15054-000, Brazil

    Valeriano A. de Oliveira

  2. Departamento de Matemática, Universidade Federal do Paraná, Setor de Ciências Exatas, Centro Politécnico, CP 019081, Jd. das Américas, Curitiba, PR, CEP 81531-990, Brazil

    Lucelina B. dos Santos

  3. Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012, Sevilla, Spain

    Rafaela Osuna-Gómez

  4. GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Casilla 447, Chillán, Chile

    Marko A. Rojas-Medar

Authors
  1. Valeriano A. de Oliveira
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  2. Lucelina B. dos Santos
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  3. Rafaela Osuna-Gómez
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  4. Marko A. Rojas-Medar
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Correspondence to Valeriano A. de Oliveira.

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de Oliveira, V.A., dos Santos, L.B., Osuna-Gómez, R. et al. Optimality conditions for nonlinear infinite programming problems. Optim Lett 9, 1131–1147 (2015). https://doi.org/10.1007/s11590-014-0808-9

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