Characterizations of Full Stability in Constrained Optimization

Abstract

This paper is mainly devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from both viewpoints of optimization theory and its applications. Based on secondorder generalized differential tools of variational analysis, we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems including problems of composite optimization, mathematical programs with polyhedral constraints as well as problems of extended and classical nonlinear programming with twice continuously differentiable data.Key words. variational analysis, constrained parametric optimization, nonlinear and extended nonlinear programming, full stability of local minimizers, strong regularity, second-order subdifferentials, parametric proxregularity and amenability AMS subject classifications. 49J52, 90C30, 90C31 Abbreviated title. Full stability in optimization

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

“…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.…”

“…Condition (3.6) can be treated as a proper extension of the classical strong second-order sufficient condition [24] to which (3.6) reduces in the case of Θ = R l − , i.e., in the case of standard equality and inequality constraints as in nonlinear programming. We refer the reader to [14,15,17,18,19] for constructive versions of (3.6) in other constraint systems. Note that (3.6) is satisfied when g is Θ-convex, i.e., the set (y, z) ∈ R m × R l g(y) − z ∈ Θ is convex.…”

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

“…Based on the new second-order calculus rules specially developed for the later construction, precise calculations of it in the settings of interest for the corresponding optimization problems as well as other variational techniques, a number of effective characterizations of Lipschitzian full stability of local minimizers have been recently obtained for several remarkable classes in finite-dimensional constrained optimization, namely for NLP, mathematical programs with polyhedral constraints, and problems of the so-called extended nonlinear programming in Mordukhovich, Rockafellar, and Sarabi (2013); for second-order cone programming in ; for semidefinite programming in ; and for minimax optimization problems in Mordukhovich and Sarabi (2014). All these characterizations, given entirely in terms of the problem data, are established under the corresponding nondegeneracy condition, which are counterparts of LICQ for the aforementioned classes of constrained optimization.…”

Approximations and solution estimates in optimization

“…There is an extensive literature on local stability of optimization problems under parametric perturbations; see for example [13,16,20,19,21,18,15] for a small collection of references. In contrast to these local stability results, dealing with "small" perturbations of an optimization problem, we present global results.…”