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3Kruger, Alexander
2López, Marco
2Mordukhovich, Boris
2Outrata, Jiri
2Van Ngai, Huynh
1Burachik, Regina
1Correa, Rafael
1Dinh, Nguyen
1Goberna, Miguel
1Hantoute, Abderrahim
1Kasimbeyli, Refail
1Khanh, Phan
1Li, Duan
1Mammadov, Musa
1Mo, T. H.
1Ramírez, Hector
1Roshchina, Vera
1Rubinov, Alex
1Sarabi, Ebrahim
1Thao, Nguyen

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90103 Numerical and Computational Mathematics
80102 Applied Mathematics
3Optimality conditions
2Error bounds
2Subdifferential
2Variational analysis
10802 Computation Theory and Mathematics
1Abstract convexity
1Augmented Lagrangian
1Coderivatives
1Computer science
1Conic programming
1Convexity
1Directional derivative
1Ekeland variational principle
1Exact penalty representation
1Facially exposed cone
1Farkas lemma
1Fenchel and approximate subdifferentials
1Full stability of local minimizers

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Stability of error bounds for semi-infinite convex constraint systems

- Van Ngai, Huynh, Kruger, Alexander, Théra, Michel

**Authors:**Van Ngai, Huynh , Kruger, Alexander , Théra, Michel**Date:**2010**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 20, no. 4 (2010), p. 2080-2096**Full Text:****Reviewed:****Description:**In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its "small" perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error bounds for semi-infinite systems of convex inequalities. By applying these characterizations, we extend some results established by AzÃ© and Corvellec [SIAM J. Optim., 12 (2002), pp. 913-927] on the sensitivity analysis of Hoffman constants to semi-infinite linear constraint systems. Copyright Â© 2010, Society for Industrial and Applied Mathematics.

**Authors:**Van Ngai, Huynh , Kruger, Alexander , Théra, Michel**Date:**2010**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 20, no. 4 (2010), p. 2080-2096**Full Text:****Reviewed:****Description:**In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its "small" perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error bounds for semi-infinite systems of convex inequalities. By applying these characterizations, we extend some results established by AzÃ© and Corvellec [SIAM J. Optim., 12 (2002), pp. 913-927] on the sensitivity analysis of Hoffman constants to semi-infinite linear constraint systems. Copyright Â© 2010, Society for Industrial and Applied Mathematics.

Abstract convexity and augmented Lagrangians

- Burachik, Regina, Rubinov, Alex

**Authors:**Burachik, Regina , Rubinov, Alex**Date:**2007**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 18, no. 2 (2007), p. 413-436**Full Text:**false**Reviewed:****Description:**The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semiinfinite optimization. © 2007 Society for Industrial and Applied Mathematics.**Description:**C1**Description:**2003005362

On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions

- Kasimbeyli, Refail, Mammadov, Musa

**Authors:**Kasimbeyli, Refail , Mammadov, Musa**Date:**2009**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 20, no. 2 (2009), p. 841-855**Full Text:****Reviewed:****Description:**In this paper we study relations between the directional derivatives, the weak subdifferentials, and the radial epiderivatives for nonconvex real-valued functions. We generalize the well-known theorem that represents the directional derivative of a convex function as a pointwise maximum of its subgradients for the nonconvex case. Using the notion of the weak subgradient, we establish conditions that guarantee equality of the directional derivative to the pointwise supremum of weak subgradients of a nonconvex real-valued function. A similar representation is also established for the radial epiderivative of a nonconvex function. Finally the equality between the directional derivatives and the radial epiderivatives for a nonconvex function is proved. An analogue of the well-known theorem on necessary and sufficient conditions for optimality is drawn without any convexity assumptions.

**Authors:**Kasimbeyli, Refail , Mammadov, Musa**Date:**2009**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 20, no. 2 (2009), p. 841-855**Full Text:****Reviewed:****Description:**In this paper we study relations between the directional derivatives, the weak subdifferentials, and the radial epiderivatives for nonconvex real-valued functions. We generalize the well-known theorem that represents the directional derivative of a convex function as a pointwise maximum of its subgradients for the nonconvex case. Using the notion of the weak subgradient, we establish conditions that guarantee equality of the directional derivative to the pointwise supremum of weak subgradients of a nonconvex real-valued function. A similar representation is also established for the radial epiderivative of a nonconvex function. Finally the equality between the directional derivatives and the radial epiderivatives for a nonconvex function is proved. An analogue of the well-known theorem on necessary and sufficient conditions for optimality is drawn without any convexity assumptions.

Second-order variational analysis in conic programming with applications to optimality and stability

- Mordukhovich, Boris, Outrata, Jiri, Ramírez, Hector

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Ramírez, Hector**Date:**2015**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 25, no. 1 (2015), p. 76-101**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for cone-constrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming. © 2015 Society for Industrial and Applied Mathematics.

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Ramírez, Hector**Date:**2015**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 25, no. 1 (2015), p. 76-101**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a second-order generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues related to conic programming. They include verifiable conditions for isolated calmness of the considered solution maps, sharp necessary optimality conditions for a class of mathematical programs with equilibrium constraints, and characterizations of tilt-stable local minimizers for cone-constrained problems. The main results obtained in the general conic programming setting are specified for and illustrated by the second-order cone programming. © 2015 Society for Industrial and Applied Mathematics.

From the Farkas lemma to the Hahn-Banach theorem

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Mo, T. H.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 2 (2014), p. 678-701**Full Text:****Reviewed:****Description:**This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) â‰¥ 0 which are consequences of a composite convex inequality (S Â° g)(x) â‰¤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 2 (2014), p. 678-701**Full Text:****Reviewed:****Description:**This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) â‰¥ 0 which are consequences of a composite convex inequality (S Â° g)(x) â‰¤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.

Peeling off a nonconvex cover of an actual convex problem: Hidden convexity

- Wu, Zhiyou, Li, Duan, Zhang, Lian-Sheng, Yang, Xin-Min

**Authors:**Wu, Zhiyou , Li, Duan , Zhang, Lian-Sheng , Yang, Xin-Min**Date:**2007**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 18, no. 2 (2007), p. 507-536**Full Text:**false**Reviewed:****Description:**Convexity is, without a doubt, one of the most desirable features in optimization. Many optimization problems that are nonconvex in their original settings may become convex after performing certain equivalent transformations. This paper studies the conditions for such hidden convexity. More specifically, some transformation-independent sufficient conditions have been derived for identifying hidden convexity. The derived sufficient conditions are readily verifiable for quadratic optimization problems. The global minimizer of a hidden convex programming problem can be identified using a local search algorithm. © 2007 Society for Industrial and Applied Mathematics.**Description:**C1**Description:**2003005616

Full stability of locally optimal solutions in second-order cone programs

- Mordukhovich, Boris, Outrata, Jiri, Sarabi, Ebrahim

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Sarabi, Ebrahim**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 4 (2014), p. 1581-1613**Full Text:****Reviewed:****Description:**The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Sarabi, Ebrahim**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 4 (2014), p. 1581-1613**Full Text:****Reviewed:****Description:**The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.

Stability of error bounds for convex constraints systems in Banach spaces

- Thera, Michel, Van Ngai, Huynh, Kruger, Alexander

**Authors:**Thera, Michel , Van Ngai, Huynh , Kruger, Alexander**Date:**2010**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 20, no. 6 (2010), p. 3280-3296**Full Text:**false**Reviewed:****Description:**This paper studies stability of error bounds for convex constraints in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered.**Description:**C1

**Authors:**Thera, Michel , Van Ngai, Huynh , Kruger, Alexander**Date:**2010**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 20, no. 6 (2010), p. 3280-3296**Full Text:**false**Reviewed:****Description:**This paper studies stability of error bounds for convex constraints in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered.**Description:**C1

Facially exposed cones are not always nice

**Authors:**Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 257-268**Full Text:****Reviewed:****Description:**We address the conjecture proposed by GÃ¡bor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case; however, there exists a four-dimensional counterexample of a cone that is facially exposed but is not nice.

**Authors:**Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 257-268**Full Text:****Reviewed:****Description:**We address the conjecture proposed by GÃ¡bor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case; however, there exists a four-dimensional counterexample of a cone that is facially exposed but is not nice.

An induction theorem and nonlinear regularity models

- Khanh, Phan, Kruger, Alexander, Thao, Nguyen

**Authors:**Khanh, Phan , Kruger, Alexander , Thao, Nguyen**Date:**2015**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

**Authors:**Khanh, Phan , Kruger, Alexander , Thao, Nguyen**Date:**2015**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 25, no. 4 (2015), p. 2561-2588**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**A general nonlinear regularity model for a set-valued mapping F : X x R+ paired right arrows Y, where X and Y are metric spaces, is studied using special iteration procedures, going back to Banach, Schauder, Lyusternik, and Graves. Namely, we revise the induction theorem from Khanh [J. Math. Anal. Appl., 118 (1986), pp. 519-534] and employ it to obtain basic estimates for exploring regularity/openness properties. We also show that it can serve as a substitution for the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping F : X paired right arrows Y. An application to second-order necessary optimality conditions for a nonsmooth set-valued optimization problem with mixed constraints is provided.

Towards supremum-sum subdifferential calculus free of qualification conditions

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

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