Stability of Locally Optimal Solutions

Abstract

Necessary and sufficient conditions are obtained for the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting. Properties of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the key to these results. A previous characterization of tilt stability comes out as a special case.

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

“…Its proof is based on the recent second-order characterizations of the fundamental notion of full stability in optimization introduced in [12].…”

“…This notion has been recognized as an important stability concept in optimization and has been completely characterized via various second-order conditions. We refer the reader to [12] and the recent papers [14,15,17,18,19] for such characterizations and their applications to broad classes of optimization and control problems. Now we are ready to formulate and prove the aforementioned proposition important in what follows.…”

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

“…Metric regularity is also closely related to the concept of tilt-stability, mainly studied in finite dimensions, see, e.g., [14,15,31,40,43], but recently also in infinite dimensions [36,39]. An extended concept incorporating tilt stability is that of full stability [30,37].…”

“…We remark that due to the linear dependence of the optimality conditions 0 ∈ J y (u, v) on y, the stability with respect to y can be seen as a form of tilt-stability [14,15,30,31,36,37,40,43] for saddle-point systems.…”

Rejoinder on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it

“…Specifically, at issue are tilt stability (Poliquin and Rockafellar 1998) and full stability (Levy et al 2000). These notions, originally introduced in the context of unconstrained minimization of an extended real-valued function, can be extended to constrained problems using the reformulation based on the indicator function of the feasible set.…”