Abstract
In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush–Kuhn–Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush–Kuhn–Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush–Kuhn–Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality.
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Acknowledgements
The authors thank the Editor-in-Chief and the anonymous reviewers for their careful reading of our manuscript and for their insightful comments and suggestions that helped us improve the quality of the paper. The work of A. A. Tret’yakov was supported by the Russian Foundation for Basic Research (project no. 17-07-00510), by Leading Scientific Schools (Grant No.8860.2016.1) and by the Presidium Programme (I.33 P RAS).
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Communicated by Boris T. Polyak.
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Brezhneva, O., Tret’yakov, A.A. When the Karush–Kuhn–Tucker Theorem Fails: Constraint Qualifications and Higher-Order Optimality Conditions for Degenerate Optimization Problems. J Optim Theory Appl 174, 367–387 (2017). https://doi.org/10.1007/s10957-017-1121-4
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DOI: https://doi.org/10.1007/s10957-017-1121-4