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10102 Applied Mathematics
1Abstract
1Composition operators
1Conjugate duality
1Constraint qualification
1Convex
1Convex programming
1Dual condition
1Functions
1Hidden convexity
1Maximal monotonicity
1Maximality of sum of two maximal monotone operators
1Polyhedral functions
1Semi-definite programming (SDP)
1Sublinear functions

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A dual criterion for maximal monotonicity of composition operators

- Jeyakumar, Vaithilingam, Wu, Zhiyou

**Authors:**Jeyakumar, Vaithilingam , Wu, Zhiyou**Date:**2007**Type:**Text , Journal article**Relation:**Set-Valued Analysis Vol. 15, no. 3 (2007), p. 265-273**Full Text:**false**Reviewed:****Description:**In this paper we present a dual criterion for the maximal monotonicity of the composition operator T:=A* SA, where S:Y→→ Y is a maximal monotone (set-valued) operator and A: X→ Y is a continuous linear map with the adjoint A*, X and Y are reflexive Banach spaces, and the product notation indicates composition. The dual criterion is expressed in terms of the closure condition involving the epigraph of the conjugate of Fitzpatrick function associated with S, and the operator A. As an easy application, a dual criterion for the maximality of the sum of two maximal monotone operators is also given. © 2006 Springer Science+Business Media B.V.**Description:**C1

Necessary and sufficient conditions for stable conjugate duality

- Burachik, Regina, Jeyakumar, Vaithilingam, Wu, Zhiyou

**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**Full Text:****Reviewed:****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

**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**Full Text:****Reviewed:****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

Hidden abstract convex functions

- Rubinov, Alex, Wu, Zhiyou, Li, Duan

**Authors:**Rubinov, Alex , Wu, Zhiyou , Li, Duan**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Nonlinear and Convex Analysis Vol. 6, no. 1 (2005), p. 203-216**Full Text:**false**Reviewed:****Description:**C1**Description:**2003001424

Sufficient conditions for global optimality of semidefinite optimization

- Quan, Jing, Wu, Zhiyou, Li, Guoquan, Wu, Ou

**Authors:**Quan, Jing , Wu, Zhiyou , Li, Guoquan , Wu, Ou**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Inequalities and Applications Vol. 2012, no. 108**Full Text:****Reviewed:****Description:**In this article, by using the Lagrangian function, we investigate the sufficient global optimality conditions for a class of semi-definite optimization problems, where the objective function are general nonlinear, the variables are mixed integers subject to linear matrix inequalities (LMIs) constraints as well as bounded constraints. In addition, the sufficient global optimality conditions for general nonlinear programming problems are derived, where the variables satisfy LMIs constraints and box constraints or bivalent constraints. Furthermore, we give the sufficient global optimality conditions for standard semi-definite programming problem, where the objective function is linear, the variables satisfy linear inequalities constraints and box constraints. © 2012 Quan et al.

**Authors:**Quan, Jing , Wu, Zhiyou , Li, Guoquan , Wu, Ou**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Inequalities and Applications Vol. 2012, no. 108**Full Text:****Reviewed:****Description:**In this article, by using the Lagrangian function, we investigate the sufficient global optimality conditions for a class of semi-definite optimization problems, where the objective function are general nonlinear, the variables are mixed integers subject to linear matrix inequalities (LMIs) constraints as well as bounded constraints. In addition, the sufficient global optimality conditions for general nonlinear programming problems are derived, where the variables satisfy LMIs constraints and box constraints or bivalent constraints. Furthermore, we give the sufficient global optimality conditions for standard semi-definite programming problem, where the objective function is linear, the variables satisfy linear inequalities constraints and box constraints. © 2012 Quan et al.

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