Optimality conditions in nonconvex optimization via weak subdifferentials
- Authors: Kasimbeyli, Refail , Mammadov, Musa
- Date: 2011
- Type: Text , Journal article
- Relation: Nonlinear Analysis, Theory, Methods and Applications Vol. 74, no. 7 (2011), p. 2534-2547
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- Description: In this paper we study optimality conditions for optimization problems described by a special class of directionally differentiable functions. The well-known necessary and sufficient optimality condition of nonsmooth convex optimization, given in the form of variational inequality, is generalized to the nonconvex case by using the notion of weak subdifferentials. The equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented. © 2011 Elsevier Ltd. All rights reserved.
Necessary and sufficient conditions for stable conjugate duality
- Authors: Burachik, Regina , Jeyakumar, Vaithilingam , Wu, Zhiyou
- Date: 2006
- Type: Text , Journal article
- Relation: Journal of Nonlinear Analysis Vol. 64, no. 9 (2006), p. 1998-2005
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- Description: The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ' −φ∗(0,v), whenever a regularity condition on φ is satisfied, is a key result in convex anal¬ysis and optimization, where φ : X × Y → IR ∪{+∞} is a convex function, X and Y are Banach spaces, Y ' is the continuous dual space of Y and φ∗ is the Fenchel-Moreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, ∗ ∗ ∈ X' inf {φ(x, 0) + x ∗(x)} = max {−φ ∗(−x ,v)}, ∀x, x∈Xv∈Y ' and obtain a new global dual regularity condition, which is much more general than the popularly known interior-point type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel-Rockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interior-point conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.
- Description: C1
- Description: 2003003596
The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus
- Authors: Baier, Robert , Farkhi, Elza , Roschina, Vera
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Theory, Methods & Applications Vol. 75, no. 3 (2012), p. 1058-1073
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- Description: We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel–Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.
Error bounds for vector-valued functions : Necessary and sufficient conditions
- Authors: Bednarczuk, Ewa , Kruger, Alexander
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis, Theory, Methods and Applications Vol. 75, no. 3 (2012), p. 1124-1140
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new derivative-like objects (slopes and subdifferentials) are introduced and a general classification scheme of error bound criteria is presented.
- Description: In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new derivative-like objects (slopes and subdifferentials) are introduced and a general classification scheme of error bound criteria is presented. © 2011 Elsevier Ltd. All rights reserved.
On coderivatives and Lipschitzian properties of the dual pair in optimization
- Authors: López, Marco , Ridolfi, Andrea , Vera De Serio, Virginia
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis, Theory, Methods and Applications Vol. 75, no. 3 (2012), p. 1461-1482
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- Description: In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem. © 2011 Elsevier Ltd. All rights reserved.
The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples
- Authors: Baier, Robert , Farkhi, Elza , Roschina, Vera
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Theory, Methods Applications Vol. 75, no. 3 (2012), p. 1074-1088
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- Description: We extend the definition of the directed subdifferential, originally introduced in [R. Baier, E. Farkhi, The directed subdifferential of DC functions, in: A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, A.J. Zaslavski (Eds.), Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe’s 70th and Simeon Reich’s 60th Birthdays, June 18–24, 2008, Haifa, Israel, in: AMS Contemp. Math., vol. 513, AMS, Bar-Ilan University, 2010, pp. 27–43], for differences of convex functions (DC) to the wider class of quasidifferentiable functions. Such generalization efficiently captures differential properties of a wide class of functions including amenable and lower/upper-View the MathML source functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: non-uniqueness and “inflation in size” of the two convex sets representing the quasidifferential after applying calculus rules. The Rubinov subdifferential is defined as the visualization of the directed subdifferential.
Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions
- Authors: Adly, Samir , Hantoute, Abderrahim , Théra, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Theory, Methods & Applications Vol. 75, no. 3 (February, 2012), p. 985-1008
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- Description: The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.