Second-order subdifferentials and convexity of real-valued functions

Chiêu, Nguyễn Huy; Huy, Nguyễn Quang

doi:10.1016/j.na.2010.08.029

Cited by 16 publications

(10 citation statements)

References 14 publications

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“…These constructions and their modification have been widely used in second-order variational analysis and its applications to stability issues and necessary optimality conditions for various classes of optimization-related and optimal control problems as well as for parametric variational and equilibrium systems. We refer the reader to [13,14,16,19,24] for the original motivations and underlying results involving the second-order subdifferentials and also to [2,3,4,5,8,9,10,11,12,17,18,20,21,22,23,28] and the bibliographies therein for more recent studies and applications.…”

Section: Introductionmentioning

confidence: 99%

Generalized differentiation of piecewise linear functions in second-order variational analysis

Mordukhovich

Sarabi

2016

Nonlinear Analysis

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The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to optimization and stability. This class consists of lower semicontinuous functions with possibly infinite values on finite-dimensional spaces, which are labeled as "piecewise linear" ones and can be equivalently described via the convexity of their epigraphs. In this the paper we calculate the second-order subdifferentials (generalized Hessians) of arbitrary convex piecewise linear functions, together with the corresponding geometric objects, entirely in terms of their initial data. The obtained formulas allow us, in particular, to justify a new exact (equality-type) second-order sum rule for such functions in the general nonsmooth setting.

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Section: Introductionmentioning

confidence: 99%

Generalized differentiation of piecewise linear functions in second-order variational analysis

Mordukhovich

Sarabi

2016

Nonlinear Analysis

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“…For C 2 functions, the latter criterion reduces to the positive-definiteness of the classical Hessian matrix-a well-known sufficient condition for the standard optimality in unconstrained problems, which happens to be necessary and sufficient for tilt-stable local minimizers [39]. We also refer the reader to the recent papers by Chieu et al [4,5] providing complete characterizations of convexity and strong convexity of nonsmooth (in the second order) functions via positive-semidefiniteness and definiteness of their second-order subdifferentials ∂ 2 ϕ. Related characterizations of monotonicity and submonotonicity of continuous mappings can be found in [6].…”

Section: Introductionmentioning

confidence: 99%

Second-Order Subdifferential Calculus with Applications to Tilt Stability in Optimization

Mordukhovich¹,

Rockafellar²

2012

SIAM J. Optim.

111

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Abstract. The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.

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“…The limiting second-order subdifferential has many applications in stability analysis of optimization problems; see, e.g., [21,22,25] and the references therein. As shown in [5,6], the Fréchet second-order subdifferential is very useful in characterizing convexity of extended-real-valued functions. The authors of [7] have shown that the Fréchet second-order subdifferential is suitable for presenting second-order necessary optimality conditions [7, Theorems 3.1 and 3.3], while the limiting second-order subdifferential works well for second-order sufficient optimality conditions [7,Theorem 4.7 and Corollary 4.8].…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions based on the Fréchet second-order subdifferential

An¹,

Yen²

2020

Preprint

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This paper focuses on second-order necessary optimality conditions for constrained optimization problems on Banach spaces. For problems in the classical setting, where the objective function is C 2 -smooth, we show that strengthened second-order necessary optimality conditions are valid if the constraint set is generalized polyhedral convex. For problems in a new setting, where the objective function is just assumed to be C 1 -smooth and the constraint set is generalized polyhedral convex, we establish sharp second-order necessary optimality conditions based on the Fréchet second-order subdifferential of the objective function and the second-order tangent set to the constraint set. Three examples are given to show that the used hypotheses are essential for the new theorems. Our second-order necessary optimality conditions refine and extend several existing results.Keywords Constrained optimization problems on Banach spaces • secondorder necessary optimality conditions • Fréchet second-order subdifferential • second-order tangent set • generalized polyhedral convex set.

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Second-order subdifferentials and convexity of real-valued functions

Cited by 16 publications

References 14 publications

Generalized differentiation of piecewise linear functions in second-order variational analysis

Generalized differentiation of piecewise linear functions in second-order variational analysis

Second-Order Subdifferential Calculus with Applications to Tilt Stability in Optimization

Optimality conditions based on the Fréchet second-order subdifferential

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