Abstract
In this article we study approximate optimality in the setting of a Banach space. We study various solution concepts existing in the literature and develop very general necessary optimality conditions in terms of limiting subdifferentials. We also study saddle point conditions and relate them to various solution concepts.
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Bustos M. (1989). Solution Approchées de Problèmes de Calcul des Variations.Revista de Matemáticas Aplicadas 10, 95–113.
Bustos M. (1994). ɛ-Gradients pour les Fonctions Localements Lipschitziennes et Applications.Numerical Functional Analysis and Applications 15, 435–453.
Clarke F.H. (1983).Optimization and Nonsmooth Analysis. Wiley-Interscience.
Hamel A. (2001). An ε-Lagrange Multiplier Rule for a Mathematical Programming Problem on Banach Spaces.Optimization 49, 137–149.
Hiriart-Urruty J.-B. (1982). ɛ-Subdifferential Calculus. In: Aubin, J.-P and Vinter R.B. (eds.),Convex Analysis and Optimization, Research Notes in Mathematics Series 57. Pitman, 43–92.
Huang X.X. and Yang X.Q. (2001). Approximate Optimal Solutions and Nonlinear Lagrange Functions.Journal of Global Optimization 21, 51–65.
Jahn J. (1996).Introduction to the Theory of Nonlinear Optimization. Springer Verlag.
Jofre A., Luc D. T. and Thera M. (1996). ɛ-Subdifferential Calculus for Nonconvex Functions and ɛ-Monotonicity.Comptes Rendus de l’Académie des Sciences 323, 735–740.
Loridan P. (1982). Necessary Conditions for ɛ-Optimality.Mathematical Programming Study 19, 140–152.
Loridan P. and Morgan, J. (1983). Penalty Functions in ɛ-Programming and ɛ-Minimax Problems.Mathematical Programming 26, 213–231.
Liu J. C. (1991). ɛ-Duality Theorems for Non-Differentiable, Non-Convex Multi-objective Programming.Journal of Optimization Theory and Applications 69, 153–167.
Mordukhovich B. (1985). On Necessary Conditions for an Extremum in Nonsmooth Optimization.Soviet Math Doklady 32, 215–220.
Mordukhovich B. (1994). Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings.Journal of Mathematical Analysis and Applications 183, 250–288.
Mordukhovich B. and Shao Y. (1996). Nonsmooth sequential Analysis in Asplund Spaces.Transactions of American Mathematical Society 348, 1235–1280.
Mordukhovich B. and Wang B. (2002). Necessary Suboptimality and Optimality Conditions via Variational Principles.SIAM Journal of Control and Optimization 41, 623–640.
Phelps R.R. (1993).Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, 1364. Springer Verlag.
Rockafellar R.T. and Wets R.J.B. (1998).Variational Analysis. Springer Verlag.
Strodiot J.-J., Nguyen V.H. and Heukemes N. (1983). ɛ-Optimal Solutions in Nondifferentiable Convex Programming and Some Related Questions.Mathematical Programming 25, 307–328.
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Part of this research was carried out when the author was a post-doctoral fellow at UAB, Barcelona by the Grant No. SB99-B0771103B of the Spanish Ministry of Education and Culture. The hospitality and the facilities provided at CODE, UAB is gratefully acknowledged.
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Dutta, J. Necessary optimality conditions and saddle points for approximate optimization in banach spaces. Top 13, 127–143 (2005). https://doi.org/10.1007/BF02578991
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DOI: https://doi.org/10.1007/BF02578991