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33Bagirov, Adil
23Rubinov, Alex
19Kruger, Alexander
17Gao, David
14López, Marco
13Ugon, Julien
13Wu, Zhiyou
10Mammadov, Musa
10Roshchina, Vera
7Outrata, Jiri
6Burachik, Regina
6Goberna, Miguel
6Sukhorukova, Nadezda
5Bai, Fusheng
5Dinh, Nguyen
5Miller, Mirka
5Thera, Michel
5Weber, Gerhard-Wilhelm
4Cánovas, Maria
4Gfrerer, Helmut

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1220102 Applied Mathematics
340802 Computation Theory and Mathematics
24Global optimization
220906 Electrical and Electronic Engineering
22Nonsmooth optimization
14Subdifferential
11Nonconvex optimization
9Metric regularity
7Normal cone
7Optimisation
7Optimization
7Problem solving
6Algorithms
6Canonical duality theory
6Constrained optimization
50101 Pure Mathematics
5Asplund space
5Augmented Lagrangian
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Abstract convexity for nonconvex optimization duality

- Nedic, A., Ozdaglar, A., Rubinov, Alex

**Authors:**Nedic, A. , Ozdaglar, A. , Rubinov, Alex**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Vol. 56, no. 5-6 (2007), p. 655-674**Full Text:**false**Reviewed:****Description:**In this article, we use abstract convexity results to study augmented dual problems for (nonconvex) constrained optimization problems. We consider a nonincreasing function f that is lower semicontinuous at 0 and establish its abstract convexity at 0 with respect to a set of elementary functions defined by nonconvex augmenting functions. We consider three different classes of augmenting functions: nonnegative augmenting functions, bounded-below augmenting functions, and unbounded augmenting functions. We use the abstract convexity results to study augmented optimization duality without imposing boundedness assumptions.**Description:**C1

Directional Holder metric regularity

- Ngai, Huynh Van, Tron, Nguyen Huu, Thera, Michel

**Authors:**Ngai, Huynh Van , Tron, Nguyen Huu , Thera, Michel**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 785-819**Full Text:****Reviewed:****Description:**This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

**Authors:**Ngai, Huynh Van , Tron, Nguyen Huu , Thera, Michel**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 785-819**Full Text:****Reviewed:****Description:**This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

Directional metric regularity of multifunctions

- Ngai, Huynh Van, Thera, Michel

**Authors:**Ngai, Huynh Van , Thera, Michel**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.**Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.

**Authors:**Ngai, Huynh Van , Thera, Michel**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.**Description:**In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.

A heuristic algorithm for solving the minimum sum-of-squares clustering problems

**Authors:**Ordin, Burak , Bagirov, Adil**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 61, no. 2 (2015), p. 341-361**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.

Spline regression models for complex multi-modal regulatory networks

- Ozmen, Ayse, Kropat, Erik, Weber, Gerhard-Wilhelm

**Authors:**Ozmen, Ayse , Kropat, Erik , Weber, Gerhard-Wilhelm**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 29, no. 3 (2014), p. 515-534**Full Text:**false**Reviewed:****Description:**Complex regulatory networks often have to be further expanded and improved with regard to the unknown effects of additional parameters and factors that can emit a disturbing influence on the key variables under consideration. The concept of target-environment (TE) networks provides a holistic framework for the analysis of such parameter-dependent multi-modal systems. In this study, we consider time-discrete TE regulatory systems with spline entries. We introduce a new regression model for these particular two-modal systems that allows us to determine the unknown system parameters by applying the multivariate adaptive regression spline (MARS) technique and the newly developed conic multivariate adaptive regression spline (CMARS) method. We obtain a relaxation by means of continuous optimization, especially, conic quadratic programming (CQP) that could be conducted by interior point methods. Finally, a numerical example demonstrates the efficiency of the spline-based approach.

A complementarity partition theorem for multifold conic systems

- Peña, Javier, Roshchina, Vera

**Authors:**Peña, Javier , Roshchina, Vera**Date:**2012**Type:**Text , Journal article**Relation:**Mathematical Programming Vol.142 , no.1-2 (2012), p.579-589**Full Text:**false**Reviewed:****Description:**Consider a homogeneous multifold convex conic system {Mathematical expression}and its alternative system {Mathematical expression}, where K 1,..., K r are regular closed convex cones. We show that there is a canonical partition of the index set {1,..., r} determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming. © 2012 Springer and Mathematical Optimization Society.

Some preconditioners for systems of linear inequalities

- Peña, Javier, Roshchina, Vera, Soheili, Negar

**Authors:**Peña, Javier , Roshchina, Vera , Soheili, Negar**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 8, no. 7 (2014), p. 2145-2152**Full Text:**false**Reviewed:****Description:**We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities. © 2014, Springer-Verlag Berlin Heidelberg.

On graphs of maximum degree 3 and defect 4

- Pineda-Villavicencio, Guillermo, Miller, Mirka

**Authors:**Pineda-Villavicencio, Guillermo , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008), p. 25-31**Full Text:**false**Reviewed:****Description:**It is well known that apart from the Petersen graph there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and

New largest known graphs of diameter 6

- Pineda-Villavicencio, Guillermo, Gómez, José, Miller, Mirka, Pérez-Rosés, Hebert

**Authors:**Pineda-Villavicencio, Guillermo , Gómez, José , Miller, Mirka , Pérez-Rosés, Hebert**Date:**2009**Type:**Text , Journal article**Relation:**Networks Vol. 53, no. 4 (2009), p. 315-328**Full Text:****Reviewed:****Description:**In the pursuit of obtaining largest graphs of given maximum degree**Description:**2003007890

**Authors:**Pineda-Villavicencio, Guillermo , Gómez, José , Miller, Mirka , Pérez-Rosés, Hebert**Date:**2009**Type:**Text , Journal article**Relation:**Networks Vol. 53, no. 4 (2009), p. 315-328**Full Text:****Reviewed:****Description:**In the pursuit of obtaining largest graphs of given maximum degree**Description:**2003007890

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.

The Demyanov–Ryabova conjecture is false

**Authors:**Roshchina, Vera**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 1 (2019), p. 227-234**Full Text:****Reviewed:****Description:**It was conjectured by Demyanov and Ryabova (Discrete Contin Dyn Syst 31(4):1273–1292, 2011) that the minimal cycle in the sequence obtained via repeated application of the Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4.

**Authors:**Roshchina, Vera**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 1 (2019), p. 227-234**Full Text:****Reviewed:****Description:**It was conjectured by Demyanov and Ryabova (Discrete Contin Dyn Syst 31(4):1273–1292, 2011) that the minimal cycle in the sequence obtained via repeated application of the Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4.

Global solutions to fractional programming problem with ratio of nonconvex functions

**Authors:**Ruan, Ning , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Applied Mathematics and Computation Vol. 255, no. (2015), p. 66-72**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for minimizing a sum of quadratic function and a ratio of nonconvex functions in Rn. By introducing a parameter, the problem is first equivalently reformed as a nonconvex polynomial minimization with elliptic constraint. It is proved that under certain conditions, the canonical dual is a concave maximization problem in R2 that exhibits no duality gap. Therefore, the global optimal solution of the primal problem can be obtained by solving the canonical dual problem. © 2014 Elsevier Inc. All rights reserved.

Canonical dual least square method for solving general nonlinear systems of quadratic equations

**Authors:**Ruan, Ning , Gao, David**Date:**2010**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 47, no. (2010), p. 335-347**Full Text:**false**Reviewed:****Description:**This paper presents a canonical dual approach for solving general non- linear algebraic systems. By using least square method, the nonlinear system of m -quadratic equations in n -dimensional space is first formulated as a nonconvex opti- mization problem. We then proved that, by the canonical duality theory developed by the second author, this nonconvex problem is equivalent to a concave maximization problem in R, which can be solved easily by well-developed convex optimization techniques. Both existence and uniqueness of global optimal solutions are discussed, and several illustrative examples are presented.**Description:**C1

The nonlinear and augmented Lagrangians for nonconvex optimization problems with a single constraint

- Rubinov, Alex, Gasimov, Rafail

**Authors:**Rubinov, Alex , Gasimov, Rafail**Date:**2002**Type:**Text , Journal article**Relation:**Applied and Computational Mathematics Vol. 1, no. 2 (2002), p. 142-157**Full Text:**false**Reviewed:****Description:**The paper contains the survey of some recent results obtained by the authors and their colleagues. We study zero duality gap properties for optimization problems with a single constraint with respect to a nonlinear penalization. The penalty function is constructed as a convolution of the objective function and the constraint by means of IPH (increasing positively homogeneous) functions. The main results are obtained for penalization by strictly IPH functions. We also examine augmented Lagrangians for optimization problems with a single constraint. We establish some links between augmented Lagrangians and Lagrange-type functions and propose a new kind of Lagrange-type functions for the problems with a single inequality constraint.**Description:**C1**Description:**2003000115

Methods for global optimization of nonsmooth functions with applications

**Authors:**Rubinov, Alex**Date:**2006**Type:**Text , Journal article**Relation:**Applied and Computational Mathematics Vol. 5, no. 1 (2006), p. 3-15**Full Text:**false**Reviewed:****Description:**In this survey paper we present some results obtained in the Centre for Informatics and Applied Optimization (CIAO) at University of Ballarat, Australia, in the area of numerical global optimization. We describe a conceptual scheme of two methods developed in CIAO and present results of numerical experiments with some real world problems. The paper is based on a plenary lecture given by the author at the First International Conference on Control and Optimization with Industrial Applications, Baku, Azerbaijan, 2005.**Description:**C1**Description:**2003001547

- Rubinov, Alex, Briec, W., Horvath, C. D

**Authors:**Rubinov, Alex , Briec, W. , Horvath, C. D**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 1, no. 1 (2005), p. 13-30**Full Text:**false**Reviewed:****Description:**A subset B of R**Description:**C1**Description:**2003001426

On the absence of duality gap for Lagrange-type functions

- Rubinov, Alex, Burachik, Regina

**Authors:**Rubinov, Alex , Burachik, Regina**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 1, no. 1 (2005), p. 33-38**Full Text:**false**Reviewed:****Description:**Given a generic dual program we discuss the absence of duality gap for a family of Lagrange-type functions. We obtain necessary conditions that become sufficient ones under some additional assumptions. We also give examples of Lagrangetype functions for which this sufficient conditions hold.**Description:**C1**Description:**2003001425

Separation properties via connectedness of topological convexity spaces

**Authors:**Sharikov, Evgeny**Date:**2010**Type:**Text , Journal article**Relation:**Pacific Journal of Optimization Vol. 6, no. 2, Suppl. 1 (2010), p. 227-241**Full Text:**false**Description:**For a given collection H of subsets of a set X we examine the convexity on X generated by H. We use a special type of connectedness of H and X for investigation of separation of convex sets by elements of H. In particular, we give a description of convex sets, which can be represented as the intersection of a subfamily of H. As an application, we give a description of abstract convex functions and sets. We also describe the abstract convex hull of a finite union of abstract convex sets.

New constructions of A-magic graphs using labeling matrices

- Sugeng, Kiki Ariyanti, Miller, Mirka

**Authors:**Sugeng, Kiki Ariyanti , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008), p. 147-151**Full Text:**false**Reviewed:**

On consecutive edge magic total labeling of graphs

- Sugeng, Kiki Ariyanti, Miller, Mirka

**Authors:**Sugeng, Kiki Ariyanti , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Discrete Algorithms Vol. 6, no. 1 (2008), p. 59-65**Full Text:**false**Reviewed:****Description:**Let G = (V, E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection**Description:**C1

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