Full Stability in Finite-Dimensional Optimization

Mordukhovich, Boris S.; Nghia, Tran T. A.; Rockafellar, R. T.

doi:10.1287/moor.2014.0669

Cited by 43 publications

(43 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

Section: Directional Derivatives Of Projection Operatorsmentioning

confidence: 99%

“…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.…”

Section: Directional Derivatives Of Projection Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

Mordukhovich

Outrata

Ramírez

2015

Set-Valued Var. Anal

View full text Add to dashboard Cite

The paper concerns parameterized equilibria governed by generalized equations whose multivalued parts are modeled via regular normals to nonconvex conic constraints. Our main goal is to derive a precise pointwise second-order formula for calculating the graphical derivative of the solution maps to such generalized equations that involves Lagrange multipliers of the corresponding KKT systems and critical cone directions. Then we apply the obtained formula to characterizing a Lipschitzian stability notion for the solution maps that is known as isolated calmness.Mathematics Subject Classification (2010): primary 49J53, 49J52; secondary 90C31.

show abstract

Section: Directional Derivatives Of Projection Operatorsmentioning

confidence: 99%

Section: Directional Derivatives Of Projection Operatorsmentioning

confidence: 99%

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

Mordukhovich

Outrata

Ramírez

2015

Set-Valued Var. Anal

View full text Add to dashboard Cite

show abstract

“…Keeping in mind that, in the current stage, the commonly used dual approach has met severe difficulties in handling tilt stability for non-polyhedral conic programs under weak conditions, examining the new approach to tilt stability for such problems would be a topic of great interest. Another important topic of further research is to expand our approach to full stability in the sense of Levy-Poliquin-Rockafellar [21], a far-going extension of tilt stability and possibly improve results developed recently in [29,30]. Furthermore, due to the strict connection of tilt stability and full stability to strong stability in the sense of Kojima [19], which is equivalent of SSOSC under MFCQ as discussed in [4,Chapter 5], studying strong stability under weaker conditions than MFCQ, e.g., MSCQ (see also our Corollary 4.12) will be an interesting topic that we will pursue.…”

Section: Discussionmentioning

confidence: 99%

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

Chiêu¹,

Hien²,

Nghia³

2018

SIAM J. Optim.

View full text Add to dashboard Cite

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...

show abstract

“…These fundamental stability concepts were introduced in optimization theory by Rockafellar and his collaborators [15,16] and then have been intensively studied by many researchers, especially in the recent years, for various classes of optimization problems; see, e.g. [7,8,[16][17][18][19][20][21][22][23][24][25][26][27] and the references therein. The construction of the second-order subdifferential/generalized Hessian in the sense of Mordukhovich [28] (i.e.…”

Section: 4)mentioning

confidence: 99%

Variational analysis of circular cone programs

2014

View full text Add to dashboard Cite

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/coderivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.

show abstract

Full Stability in Finite-Dimensional Optimization

Cited by 43 publications

References 37 publications

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

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

Variational analysis of circular cone programs

Contact Info

Product

Resources

About