Critical multipliers in variational systems via second-order generalized differentiation

Abstract

Abstract. In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush-Kuhn-Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitzlike/Aubin property of solution maps yields their robust isolated calmness.

Criticality of Lagrange multipliers in extended nonlinear optimization

“…The main results of the current paper extend those from [15], obtained for CPWL functions θ, to the case of functions θ Y,B defined in (1.2), which form a major class of extended-real-valued convex piecewise linear-quadratic functions in variational analysis; see [18] and Section 2 below. At the same time, the new results obtained here are completely independent from those derived for the variational system (1.4) in [15] in the case of nonpolyhedral sets Θ therein.…”

“…).where f and Φ are the same as in (1.3), and where N Θ is the normal cone to a C 2 -cone reducible set Θ ⊂ R m . This framework covers, in particular, KKT systems associated with general problems of (nonpolyhedral) conic programming; see, e.g., [1].The main results of the current paper extend those from [15], obtained for CPWL functions θ, to the case of functions θ Y,B defined in (1.2), which form a major class of extended-real-valued convex piecewise linear-quadratic functions in variational analysis; see [18] and Section 2 below. At the same time, the new results obtained here are completely independent from those derived for the variational system (1.4) in [15] in the case of nonpolyhedral sets Θ therein.…”

“…In [15], Mordukhovich and Sarabi suggested new definitions of critical and noncritical multipliers for a general class of subdifferential variational systems of type (1.3), where θ may be even a nonconvex extended-real-valued function. The given definitions in [15] are expressed via second-order generalized differential constructions of variational analysis while reduced to those from [4,5] for the classical KKT systems corresponding to NLPs.…”

“…In [15], Mordukhovich and Sarabi suggested new definitions of critical and noncritical multipliers for a general class of subdifferential variational systems of type (1.3), where θ may be even a nonconvex extended-real-valued function. The given definitions in [15] are expressed via second-order generalized differential constructions of variational analysis while reduced to those from [4,5] for the classical KKT systems corresponding to NLPs. Furthermore, for extendedreal-valued convex piecewise linear (CPWL) functions θ in (1.3), which include (1.2) when B = 0, the definitions of critical and noncritical multipliers are expressed in [15] entirely in terms of the problem data with the subsequent characterizations of criticality and various applications to optimization and stability problems for such systems.…”

“…The quite recent paper of the same authors [16] contains counterparts of some major results from [15] with developing also novel issues on criticality for variational systems described by…”