Computation of Graphical Derivative for a Class of Normal Cone Mappings under a Very Weak Condition

Chiêu, Nguyễn Huy; Hien, Le Van

doi:10.1137/16m1066816

Cited by 16 publications

(13 citation statements)

References 29 publications

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“…The obtained major result is the first in the literature for nonpolyhedral constraint systems without imposing nondegeneracy. As discussed below, its proof is significantly different from the recent ones given in [5,8,10] for polyhedral systems, even in the latter case. It is also largely different from the approaches developed in [11,19,20] for conic programs under nondegeneracy assumptions.…”

Section: Introductionmentioning

confidence: 64%

“…(i) First we highlight some important differences between our proof of Theorem 5.1 for nonpolyhedral second-order constraint systems and its polyhedral counterpart for NLPs in [10, Theorem 1] and in the similar devices from [5,8]. Unlike the latter proof that exploits the Hoffman lemma to verify the inclusion "⊂" in (5.1), we do not appeal to any error bound estimate; this is new even for polyhedral systems.…”

Section: Graphical Derivative Of the Normal Cone Mappingmentioning

confidence: 99%

“…In Section 4 we study second-order properties of the SOCP constraint system (1.1) by using the twice epi-differentiability of δ Q and the metric subregularity constraint qualification (MSCQ) for (1.1), which seems to be the weakest constraint qualification that has been investigated and employed recently in the (polyhedral) NLP framework; see [10,8,5]. Among the most important results obtained in this section we mention the following: (i) a constructive description of generalized normals to the critical cone at the point in question under MSCQ, and (ii) a characterization of the uniqueness of Lagrange multipliers together with an appropriate error bound estimate (automatic in the polyhedral case) at stationary points via a new constraint qualification in conic programming, which happens to be in the case of (1.1) a dual form of the strict Robinson constraint qualification (SRCQ) from [4,6].…”

Section: Introductionmentioning

confidence: 99%

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Second-order variational analysis in second-order cone programming

2018

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The paper conducts a second-order variational analysis for an important class of nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone Q. From one hand, we prove that the indicator function of Q is always twice epi-differentiable and apply this result to characterizing the uniqueness of Lagrange multipliers at stationary points together with an error bound estimate in the general second-order cone setting involving C 2 -smooth data. On the other hand, we precisely calculate the graphical derivative of the normal cone mapping to Q under the weakest metric subregularity constraint qualification and then give an application of the latter result to a complete characterization of isolated calmness for perturbed variational systems associated with second-order cone programs. The obtained results seem to be the first in the literature in these directions for nonpolyhedral problems without imposing any nondegeneracy assumptions.Problems of this type are challenging mathematically while being important for various applications; see, e.g., [1,3,4,21,23,24] and the bibliographies therein. A remarkable feature of SOCPs, which significantly distinguishes them from nonlinear programs (NLPs) and the like, is the nonpolyhedrality of the underlying Lorentz cone Q in (1.1).The main intention of this paper is to conduct a second-order analysis for SOCPs by using appropriate tools of second-order generalized differentiation. The following two topics are of our particular interest here: (1) to prove the twice epi-differentiability of the indicator function

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Section: Introductionmentioning

confidence: 64%

Section: Graphical Derivative Of the Normal Cone Mappingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Second-order variational analysis in second-order cone programming

2018

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“…In this section, using our subgradient graphical derivative characterization of tilt-stability along with the recent formulas for the graphical derivative of normal cone mappings from [5,15] and some techniques from [12], we establish new results on tilt stability for nonlinear programming problems under the metric subregular constraint qualification.…”

Section: Tilt Stability In Nonlinear Programmingmentioning

confidence: 99%

“…In fact, the subgradient graphical derivative and the second-order subdifferential are generally independent concepts; however, under additional conditions, the value of the subgradient graphical derivative can be identified with a subset of the value of the second-order subdifferential; see [40,41]. We note that one of the biggest advantages of this approach is the workable computation of the graphical derivative in various important cases under very mild assumptions in initial data; see [5,14,15,33]. Furthermore, several results on tilt stability, e.g.…”

mentioning

confidence: 99%

Characterization of Tilt Stability via Subgradient Graphical Derivative with Applications to Nonlinear Programming

Chiêu¹,

Hien²,

Nghia³

2018

SIAM J. Optim.

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This paper is devoted to the study of tilt stability in finite dimensional optimization via the approach of using the subgradient graphical derivative. We establish a new characterization of tilt-stable local minimizers for a broad class of unconstrained optimization problems in terms of a uniform positive definiteness of the subgradient graphical derivative of the objective function around the point in question. By applying this result to nonlinear programming under the metric subregularity constraint qualification, we derive a second-order characterization and several new sufficient conditions for tilt stability. In particular, we show that each stationary point of a nonlinear programming problem satisfying the metric subregularity constraint qualification is a tilt-stable local minimizer if the classical strong second-order sufficient condition holds. in question. Furthermore, using this result together with a formula of Dontchev and Rockafellar [7] for the second-order limiting subdifferential of the indicator function of polyhedral convex set, they obtained a second-order characterization of tilt-stability for nonlinear programming problems with linear constraints [37, Theorem 4.5]. The main difficulty in applying the tilt-stability characterization of Poliquin and Rockafellar [37] to other nonlinear constrained optimization problems is the computation/estimation of the second-order subdifferential in terms of explicit problem data.By establishing new second-order subdifferential calculi, Mordukhovich and Rockafellar [34] derived second-order characterizations of tilt-stable minimizers for some classes of constrained optimization problems. Among other important things, they showed that for C 2 -smooth nonlinear programming problems, under the linear independence constraint qualification (LICQ), a stationary point is a tilt-stable local minimizer if and only if the strong second-order sufficient condition (SSOSC) holds. Consequently, in this setting, tiltstability is equivalent to Robinson's strong regularity [39] of the associated Karush-Kuhn-Tucker system whenever LICQ occurs at the point in question. In contrast to Robinson's strong regularity, tilt stability does not postulate LICQ as a necessary condition. This observation motivated the study of tilt-stability for nonlinear programming under constraint qualifications weaker than LICQ, aiming to cover a broader class of examined problems.Under the validity of both the Mangasarian-Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ), Mordukhovich and Outrata [31] proved that SSOSC is a sufficient condition for a stationary point to be a tilt-stable local minimizer in nonlinear programming. In [28] Mordukhovich and Nghia showed that SSOSC is indeed not a necessary condition for tilt stability and then introduced the uniform second-order sufficient condition (USOSC) to characterize tilt stability when both MFCQ and CRCQ occur. Recently, Gfrerer and Mordukhovich [12] obtained some pointbased second-order sufficient c...

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Well-Posedness and Coderivative Calculus

Mordukhovich

2018

Variational Analysis and Applications

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Computation of Graphical Derivative for a Class of Normal Cone Mappings under a Very Weak Condition

Cited by 16 publications

References 29 publications

Second-order variational analysis in second-order cone programming

Second-order variational analysis in second-order cone programming

Characterization of Tilt Stability via Subgradient Graphical Derivative with Applications to Nonlinear Programming

Well-Posedness and Coderivative Calculus

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