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When the Karush-Kuhn-Tucker theorem fails: constraint qualifications and higher-order optimality conditions for degenerate optimization problems. (English) Zbl 1377.49025

Summary: 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.

MSC:

49K27 Optimality conditions for problems in abstract spaces
49N60 Regularity of solutions in optimal control
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
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