Tilt Stability, Uniform Quadratic Growth, and Strong Metric Regularity of the Subdifferential

Lewis, Adrian S.

doi:10.1137/120876551

Cited by 82 publications

(86 citation statements)

References 12 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Approximations and solution estimates in optimization

Røyset

2017

Math. Program.

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Abstract. Approximation is central to many optimization problems and the supporting theory provides insight as well as foundation for algorithms. In this paper, we lay out a broad framework for quantifying approximations by viewing finite-and infinite-dimensional constrained minimization problems as instances of extended real-valued lower semicontinuous functions defined on a general metric space. Since the Attouch-Wets distance between such functions quantifies epi-convergence, we are able to obtain estimates of optimal solutions and optimal values through estimates of that distance. In particular, we show that near-optimal and near-feasible solutions are effectively Lipschitz continuous with modulus one in this distance. We construct a general class of approximations of extended real-valued lower semicontinuous functions that can be made arbitrarily accurate and that involve only a finite number of parameters under additional assumptions on the underlying metric space.

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Section: Introductionmentioning

confidence: 99%

Approximations and solution estimates in optimization

Røyset

2017

Math. Program.

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“…It can also be used to directly characterize the convergence of certain basic optimization methods [4,25,26]. Metric regularity is also closely related to the concept of tilt-stability, mainly studied in finite dimensions, see, e.g., [14,15,31,40,43], but recently also in infinite dimensions [36,39]. An extended concept incorporating tilt stability is that of full stability [30,37].…”

Section: V) − K(u)mentioning

confidence: 99%

“…We remark that due to the linear dependence of the optimality conditions 0 ∈ J y (u, v) on y, the stability with respect to y can be seen as a form of tilt-stability [14,15,30,31,36,37,40,43] for saddle-point systems.…”

Section: Stability With Respect To Datamentioning

confidence: 99%

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

Clason

Valkonen

2016

Set-Valued Var. Anal

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We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.

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“…In 2008, under the convexity assumption of f , Aragón Artacho and Geoffroy [1] first studied the stable second order local minimizer of f in terms of the subdifferential mapping ∂f and proved thatx ∈ dom(f ) is a stable second order local minimizer of f if and only if ∂f is strongly metrically regular at (x, 0). In 2013, under the finite dimension assumption, Drusvyatskiy and Lewis [9] extended Aragón Artacho and Geoffroy's result to the prox-regularity and subdifferential continuity case. Recently, these works have been pushed by Drusvyatskiy, Mordukhovich, Nghia and Outrata (cf.…”

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confidence: 96%

“…On the other hand, corresponding to the special case when ϕ(t) = t 2 and ψ(t) = t, Drusvyatskiy and Lewis [9] did prove that the stable second order local minimizer and tilt-stable local minimum are equivalent. Thus, it is natural to ask whether there exists an exact relationship between ϕ and ψ such that ϕ-SLWP and ψ-TSLM are equivalent.…”

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confidence: 99%

Stable well-posedness and tilt stability with respect to admissible functions

Zheng

Zhu

2017

ESAIM: COCV

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Abstract. Note that the well-posedness of a proper lower semicontinuous function f can be equivalently described using an admissible function. In the case when the objective function f undergos the tilt perturbations in the sense of Poliquin and Rockafellar, adopting admissible functions ϕ and ψ, this paper introduces and studies the stable well-posedness of f with respect to ϕ (in breif, ϕ-SLWP) and tilt-stable local minimum of f with respect to ψ (in brief, ψ-TSLM). In the special case when ϕ(t) = t 2 and ψ(t) = t, the corresponding ϕ-SLWP and ψ-TSLM reduce to the stable second order local minimizer and tilt stable local minimum respectively, which have been extensively studied in recent years. We discover an interesting relationship between two admissible functions ϕ and ψ: ψ(t) = (ϕ ′ ) −1 (t), which implies that a proper lower semicontinous function f on a Banach space has ϕ-SLWP if and only if f has ψ-TSLM. Using the techniques of variational analysis and conjugate analysis, we also prove that the strong metric ϕ ′ -regularity of ∂f is a sufficient condition for f to have ϕ-SLWP and that the strong metric ϕ ′ -regularity of ∂co(f + δ B [x,r] ) for some r > 0 is a necessary condition for f to have ϕ-SLWP. In the special case when ϕ(t) = t 2 , our results cover some existing main results on the tilt stability.

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Tilt Stability, Uniform Quadratic Growth, and Strong Metric Regularity of the Subdifferential

Abstract: We prove that uniform second order growth, tilt stability, and strong metric regularity of the limiting subdifferential -three notions that have appeared in entirely different settings -are all essentially equivalent for any lower-semicontinuous, extended-real-valued function.

Cited by 82 publications

References 12 publications

Approximations and solution estimates in optimization

Approximations and solution estimates in optimization

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

Stable well-posedness and tilt stability with respect to admissible functions

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