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Optimality conditions based on the Fréchet second-order subdifferential

Abstract

This paper focuses on second-order necessary optimality conditions for constrained optimization problems on Banach spaces. For problems in the classical setting, where the objective function is \(C^2\)-smooth, we show that strengthened second-order necessary optimality conditions are valid if the constraint set is generalized polyhedral convex. For problems in a new setting, where the objective function is just assumed to be \(C^1\)-smooth and the constraint set is generalized polyhedral convex, we establish sharp second-order necessary optimality conditions based on the Fréchet second-order subdifferential of the objective function and the second-order tangent set to the constraint set. Three examples are given to show that the used hypotheses are essential for the new theorems. Our second-order necessary optimality conditions refine and extend several existing results.

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Acknowledgements

This research was supported by Vietnam Institute for Advanced Study in Mathematics (VIASM). Duong Thi Viet An was also supported by the Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology. Helpful comments of the handling Associate Editor are gratefully acknowledged.

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Correspondence to D. T. V. An.

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An, D.T.V., Yen, N.D. Optimality conditions based on the Fréchet second-order subdifferential. J Glob Optim 81, 351–365 (2021). https://doi.org/10.1007/s10898-021-01011-4

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Keywords

  • Constrained optimization problems on Banach spaces
  • Second-order necessary optimality conditions
  • Fréchet second-order subdifferential
  • Second-order tangent set
  • Generalized polyhedral convex set

Mathematics Subject Classification

  • 49K27
  • 49J53
  • 90C30
  • 90C46
  • 90C20