Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification (Q2871610)

For inequality constrained minimization problems defined by twice continuously differentiable functions, the authors present several characterizations of tilt stability of local minimizers, in the sense of \textit{R. A. Poliquin} and \textit{R. T. Rockafellar} [SIAM J. Optim. 8, No. 2, 287--299 (1998; Zbl 0918.49016)], under the Mangasarian - Fromovitz constraint qualification. One of them is in terms of the Hessian of the objective function, while another one involves the strong metrical regularity of an associated set-valued mapping. Imposing, additionally, the Constant Rank Constraint Qualification, other coderivative conditions, one sufficient and one necessary, for tilt stability, using the Hessian of the Lagrangian, are obtained. The sufficient condition is proved to be equivalent to the positive definiteness of the Lagrangian on the orthogonal subspace to the gradients of the constraint functions corresponding to the strictly positive multipliers. Under both constraint qualifications, the authors show that tilt stability implies a calmness property of a natural perturbation of the KKT system. Some illustrative examples are presented.