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5Kruger, Alexander
2Baier, Robert
2Farkhi, Elza
2Gao, David
2Mammadov, Musa
2Outrata, Jiri
2Pineda-Villavicencio, Guillermo
2Plubtieng, Somyot
2Roschina, Vera
2Seangwattana, Thidaporn
1Adam, Lukas
1Adly, Samir
1Ali, Elaf
1Cibulka, Radek
1Delorme, Charles
1Fabian, Marian
1Feria-Purón, Ramiro
1Grigoryev, Igor
1Hantoute, Abderrahim
1Hassani, Sara

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40906 Electrical and Electronic Engineering
20103 Numerical and Computational Mathematics
2Differences of sets
2Directed sets
2Directed subdifferential
2Optimal control
2Perturbation
2Quasidifferentiable functions
2Smooth variational principle
2Subdifferentials
11502 Banking, Finance and Investment
1Billiards
1Bipartite Moore bound
1Borwein-Preiss variational principle
1Borwein–Preiss variational principle
1Canonical dual finite element method
1Canonical duality theory
1Closed-convexification

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Sigma supporting cone and optimality conditions in non-convex problems

- Hassani, Sara, Mammadov, Musa

**Authors:**Hassani, Sara , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Far East Journal of Mathematical Sciences Vol. 91, no. 2 (2014), p. 169-190**Full Text:**false**Reviewed:****Description:**In this paper, a new supporting function for characterizing non-convex sets is introduced. The notions of Ïƒ-supporting cone and maximal conic gap are proposed and some properties are investigated. By applying these new notions, we establish the optimality conditions considered in [7] for a broader class of finite dimensional normed spaces in terms of weak subdifferentials.

Quadratic form representations via generalized continuants

- Delorme, Charles, Pineda-Villavicencio, Guillermo

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Integer Sequences Vol. 18, no. 6 (2015), p. Article number 15.6.4**Full Text:**false**Reviewed:****Description:**H. J. S. Smith proved Fermat’s two-square theorem using the notion of palindromic continuants. In this paper we extend Smith’s approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper representations. © 2015 University of Waterloo. All rights reserved.

Global asymptotic stability in a class of nonlinear differential delay equations

- Ivanov, Anatoli, Mammadov, Musa

**Authors:**Ivanov, Anatoli , Mammadov, Musa**Date:**2011**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems Vol. 2011, no. Supplement 2011 (2011), p.**Full Text:****Reviewed:****Description:**An essentially nonlinear dierential equation with delay serving as a mathematical model of several applied problems is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are de- rived. An application to a physiological model by M.C. Mackey is treated in detail.**Description:**2003009358

**Authors:**Ivanov, Anatoli , Mammadov, Musa**Date:**2011**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems Vol. 2011, no. Supplement 2011 (2011), p.**Full Text:****Reviewed:****Description:**An essentially nonlinear dierential equation with delay serving as a mathematical model of several applied problems is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are de- rived. An application to a physiological model by M.C. Mackey is treated in detail.**Description:**2003009358

Density character of subgroups of topological groups

- Leiderman, Arkady, Morris, Sidney, Tkachenko, Mikhail

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

Comparative analysis of numerical solution of optimal control problems

- Shangareeva, Gulnaz, Grigoryev, Igor, Mustafina, Svetlana

**Authors:**Shangareeva, Gulnaz , Grigoryev, Igor , Mustafina, Svetlana**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Pure and Applied Mathematics Vol. 110, no. 4 (2016), p. 645-649**Full Text:****Reviewed:****Description:**In this article step by step algorithms were developed for solving optimal control problems based on the method of successive approximations and the method of variations in the space of controls. The algorithm of the method of successive approximations requires details of the problem to the boundary problem of the maximum principle. In turn, the algorithm of the variations is more versatile because it is based on iterating state variables and control in the phase space. A numerical study and comparative analysis of the developed algorithms performed at different values of accuracy. © 2016 Academic Publications, Ltd.

**Authors:**Shangareeva, Gulnaz , Grigoryev, Igor , Mustafina, Svetlana**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Pure and Applied Mathematics Vol. 110, no. 4 (2016), p. 645-649**Full Text:****Reviewed:****Description:**In this article step by step algorithms were developed for solving optimal control problems based on the method of successive approximations and the method of variations in the space of controls. The algorithm of the method of successive approximations requires details of the problem to the boundary problem of the maximum principle. In turn, the algorithm of the variations is more versatile because it is based on iterating state variables and control in the phase space. A numerical study and comparative analysis of the developed algorithms performed at different values of accuracy. © 2016 Academic Publications, Ltd.

Fractal bodies invisible in 2 and 3 directions

- Plakhov, Alexander, Roshchina, Vera

**Authors:**Plakhov, Alexander , Roshchina, Vera**Date:**2013**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series A Vol. 33, no. 4 (2013), p. 1615-1631**Full Text:****Reviewed:****Description:**We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 12], where two-dimensional bodies invisible in one direction and threedimensional bodies invisible in one and two orthogonal directions were constructed.**Description:**2003010679

**Authors:**Plakhov, Alexander , Roshchina, Vera**Date:**2013**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series A Vol. 33, no. 4 (2013), p. 1615-1631**Full Text:****Reviewed:****Description:**We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 12], where two-dimensional bodies invisible in one direction and threedimensional bodies invisible in one and two orthogonal directions were constructed.**Description:**2003010679

Borwein–Preiss vector variational principle

- Kruger, Alexander, Plubtieng, Somyot, Seangwattana, Thidaporn

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

On optimal control of a sweeping process coupled with an ordinary differential equation

**Authors:**Adam, Lukas , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series B Vol. 19, no. 9 (November 2014 2014), p. 2709-2738**Full Text:**false**Reviewed:****Description:**We study a special case of an optimal control problem governed by a differential equation and a differential rate{independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process. We perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.

Internet security applications of Grobner-Shirvov bases

- Kelarev, Andrei, Yearwood, John, Watters, Paul

**Authors:**Kelarev, Andrei , Yearwood, John , Watters, Paul**Date:**2010**Type:**Text , Journal article**Relation:**Asian-European Journal of Mathematics Vol. 3, no. 3 (2010), p. 435-442**Relation:**http://purl.org/au-research/grants/arc/DP0211866**Full Text:**false**Reviewed:**

On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

- Baier, Robert, Farkhi, Elza, Roschina, Vera

**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**Full Text:**false**Reviewed:****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.

- Baier, Robert, Farkhi, Elza, Roschina, Vera

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

- Adly, Samir, Hantoute, Abderrahim, Théra, Michel

**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**Full Text:**false**Reviewed:****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.

On SPD method for solving canonical dual problem in post buckling of large deformed elastic beam

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

On large bipartite graphs of diameter 3

- Feria-Purón, Ramiro, Miller, Mirka, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers dâ‰¥2 and Dâ‰¥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-PurÃ³n and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.**Description:**2003011037

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers dâ‰¥2 and Dâ‰¥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-PurÃ³n and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.**Description:**2003011037

**Authors:**López, Marco , Volle, Michel**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 390, no. 1 (2012), p. 307-312**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**In this paper we approach the study of the subdifferential of the closed convex hull of a function and the related integration problem without the usual assumption of epi-pointedness. The key tool is, as in Hiriart-Urruty et al. (2011) [7], the concept of ε-subdifferential. Some other assumptions which are standard in the literature are also removed.

About [q]-regularity properties of collections of sets

- Kruger, Alexander, Thao, Nguyen

**Authors:**Kruger, Alexander , Thao, Nguyen**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 416, no. 2 (2014), p. 471-496**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.**Description:**We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed. (C) 2014 Elsevier Inc. All rights reserved.

**Authors:**Kruger, Alexander , Thao, Nguyen**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 416, no. 2 (2014), p. 471-496**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.**Description:**We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed. (C) 2014 Elsevier Inc. All rights reserved.

Borwein-Preiss variational principle revisited

- Kruger, Alexander, Plubtieng, Somyot, Seangwattana, Thidaporn

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.

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.

On relaxing the Mangasarian-Fromovitz constraint qualification

- Kruger, Alexander, Minchenko, Leonld, Outrata, Jiri

**Authors:**Kruger, Alexander , Minchenko, Leonld , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Positivity Vol. 18, no. 1 (2014), p. 171-189**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**For the classical nonlinear program, two new relaxations of the Mangasarian– Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.

**Authors:**Kruger, Alexander , Minchenko, Leonld , Outrata, Jiri**Date:**2014**Type:**Text , Journal article**Relation:**Positivity Vol. 18, no. 1 (2014), p. 171-189**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**For the classical nonlinear program, two new relaxations of the Mangasarian– Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.

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