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

Mordukhovich, Boris S.

doi:10.1007/s11750-015-0370-3

Cited by 7 publications

(9 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The next theorem shows that full stability of the given locally optimal solutionx to (5.2) with θ ∈ CP W L rules out the existence of critical multipliers associated withx. This proves the conjecture of [28] for the class of composite optimization problems (1.2) studied in the paper; see Section 1 for more discussions.…”

Section: Noncriticality From Full Stability In Composite Optimizationsupporting

confidence: 75%

Critical multipliers in variational systems via second-order generalized differentiation

Mordukhovich

Sarabi

2017

Math. Program.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Noncriticality From Full Stability In Composite Optimizationsupporting

confidence: 75%

Critical multipliers in variational systems via second-order generalized differentiation

Mordukhovich

Sarabi

2017

Math. Program.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore it is highly desired from the numerical viewpoint to rule out the existence of critical multipliers and so to be able making such a conclusion based on the initial data of the NLP in question. These and related issues have been discussed in the recent comments of the second author [23] on the survey by Izmailov and Solodov devoted to critical multipliers, which is based on their book [14]. It is conjectured in [23] that under appropriate qualification conditions tilt stability excludes the existence of critical multipliers.…”

Section: Discussion and Examplesmentioning

confidence: 99%

“…A major goal of the future research is to extend the theory of tilt stability developed in this paper to the case of fully stable local minimizers in NLPs. Note that full stable minimizers seem to be more appropriate to rule out critical multipliers according to the second conjecture in [23].…”

Section: Discussion and Examplesmentioning

confidence: 99%

“…These and related issues have been discussed in the recent comments of the second author [23] on the survey by Izmailov and Solodov devoted to critical multipliers, which is based on their book [14]. It is conjectured in [23] that under appropriate qualification conditions tilt stability excludes the existence of critical multipliers.…”

Section: Theorem 78 (Pointbased Second-order Characterization Of Tilt...mentioning

confidence: 99%

See 1 more Smart Citation

Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions

Gfrerer¹,

Mordukhovich²

2015

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is devoted to the study of tilt stability of local minimizers for classical nonlinear programs with equality and inequality constraints in finite dimensions described by twice continuously differentiable functions. The importance of tilt stability has been well recognized from both theoretical and numerical perspectives of optimization, and this area of research has drawn much attention in the literature, especially in recent years. Based on advanced techniques of variational analysis and generalized differentiation, we derive here complete pointbased second-order characterizations of tilt-stable minimizers entirely in terms of the initial program data under the new qualification conditions, which are the weakest ones for the study of tilt stability.

show abstract

“…It is natural to suppose that seeking not arbitrary while just "nice" and stable in some sense local minimizers allows us to rule out the appearance of critical multipliers associated with such local optimal solutions. It is conjectured in [10] that fully stable local minimizers in the sense of [7] are appropriate candidate for excluding critical multipliers. This conjecture is affirmatively verified in [14] for problems (1.1) with θ = θ Y,B where B = 0.…”

Section: Critical Multipliers and Full Stability Of Minimizers In Enlpsmentioning

confidence: 99%

Criticality of Lagrange multipliers in extended nonlinear optimization

Mordukhovich

Sarabi

2020

Optimization

Self Cite

View full text Add to dashboard Cite

The paper is devoted to the study and applications of criticality of Lagrange multipliers in variational systems, which are associated with the class of problems in composite optimization known as extended nonlinear programming (ENLP). The importance of both ENLP and the concept of multiplier criticality in variational systems has been recognized in theoretical and numerical aspects of optimization and variational analysis, while the criticality notion has never been investigated in the ENLP framework. We present here a systematic study of critical and noncritical multipliers in a general variational setting that covers, in particular, KKT systems in ENLP with establishing their verifiable characterizations as well as relationships between noncriticality and other stability notions in variational analysis. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation. ).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.

show abstract

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

Cited by 7 publications

References 21 publications

Critical multipliers in variational systems via second-order generalized differentiation

Critical multipliers in variational systems via second-order generalized differentiation

Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions

Criticality of Lagrange multipliers in extended nonlinear optimization

Contact Info

Product

Resources

About