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Necessary Optimality Conditions for Interval Optimization Problems with Functional and Abstract Constraints

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Abstract

This work addresses interval optimization problems in which the objective function is interval-valued while the constraints are given in functional and abstract forms. The functional constraints are described by means of both inequalities and equalities. The abstract constraint is expressed through a closed and convex set with a nonempty interior. Necessary optimality conditions are derived, given in a multiplier rule structure involving the gH-gradient of the interval objective function along with the (classical) gradients of the constraint functions and the normal cone to the set related to the abstract constraint. The main tool is a specification of the Dubovitskii–Milyutin formalism. We defined an appropriated notion of directions of decrease to an interval-valued function, using the lower–upper partial ordering of the interval space (LU order).

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Acknowledgements

This work was supported by National Council for the Improvement of Higher Education—CAPES/UNESP/IBILCE (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) [Finance Code 001]; São Paulo Research Foundation—FAPESP (Fundação de Amparo à Pesquisa do Estadode São Paulo) [Grant Number 2013/07375-0]; National Council for Scientific and Technological Development—CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) [Grant Number 305786/2018-0]; and by the Dean’s Office for Graduate Studies of the São Paulo State University—PROPG (Pró-Reitoria de Pós-Graduação) [Edital 6/2021]. This work was developed while F. R. Villanueva was a Ph.D. studant at the São Paulo State University (Unesp).

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Authors and Affiliations

  1. Universidad Mayor de San Andrés, Carrera de Matemática, La Paz, Bolivia

    Fabiola Roxana Villanueva

  2. São Paulo State University, São José do Rio Preto, São Paulo, Brazil

    Valeriano Antunes de Oliveira

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  1. Fabiola Roxana Villanueva
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  2. Valeriano Antunes de Oliveira
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Correspondence to Valeriano Antunes de Oliveira.

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Communicated by Juan-Enrique Martinez Legaz.

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Villanueva, F.R., de Oliveira, V.A. Necessary Optimality Conditions for Interval Optimization Problems with Functional and Abstract Constraints. J Optim Theory Appl 194, 896–923 (2022). https://doi.org/10.1007/s10957-022-02055-6

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  • DOI: https://doi.org/10.1007/s10957-022-02055-6

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