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A new algorithm for the placement of WLAN access point based on nonsmooth optimization technique

  • Authors: Kouhbor, Shahnaz , Ugon, Julien , Kruger, Alexander , Rubinov, Alex , Branch, Philip
  • Date: 2005
  • Type: Text , Conference paper
  • Relation: Paper presented at the 7th International Conference on Advanced Communication Technology, Phoenix Park, Korea : 21st February, 2005
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  • Description: In wireless local area network (WLAN), signal coverage is obtained by proper placement of access points (APs). The impact of incorrect placement of APs is significant. If they are placed too far apart, they generate a coverage gap but if they are too close to each other, this leads to excessive co-channel interferences. In this paper, we describe a mathematical model we have developed to find the optimal number and location of APs. To solve the problem, we use an optimization algorithm developed at the University of Ballarat called discrete gradient algorithm. Results indicate that our model is able to solve optimal coverage problems for different numbers of users.
  • Description: E1
  • Description: 2003001376

About [q]-regularity properties of collections of sets

  • 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
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  • 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.

About errors bounds in metric spaces

About extensions of the extremal principle

  • Authors: Bui, Hoa , Kruger, Alexander
  • Date: 2018
  • Type: Text , Journal article
  • Relation: Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 215-242
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings. © 2018, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

About intrinsic transversality of pairs of sets

  • Authors: Kruger, Alexander
  • Date: 2018
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.

About regularity of collections of sets

  • Authors: Kruger, Alexander
  • Date: 2006
  • Type: Text , Journal article
  • Relation: Set-Valued Analysis Vol. 14, no. 2 (Jun 2006), p. 187-206
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  • Description: The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular.
  • Description: 2003001526

About stationarity and regularity in variational analysis

  • Authors: Kruger, Alexander
  • Date: 2009
  • Type: Text , Journal article
  • Relation: Taiwanese Journal of Mathematics Vol. 13, no. 6A (2009), p. 1737-1785
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  • Description: Stationarity and regularity concepts for the three typical for variational analysis classes of objects - real-valued functions, collections of sets, and multifunctions - are investigated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity properties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed.

About stationarity and regularity of real-valued functions

About subtransversality of collections of sets

About uniform regularity of collection sets

An induction theorem and nonlinear regularity models

  • 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
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  • 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.

Boris Mordukhovich, the never tiring traveller, celebrates his sixtieth birthday

Borwein-Preiss variational principle revisited

Borwein–Preiss vector variational principle

  • 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
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  • 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.

Calmness modulus of linear semi-infinite programs

  • Authors: Cánovas, Maria , Kruger, Alexander , López, Marco , Parra, Juan , Théra, Michel
  • Date: 2014
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 29-48
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
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  • Description: Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

Calmness of efficient solution maps in parametric vector optimization

  • Authors: Chuong, Thai Doan , Kruger, Alexander , Yao, J. C.
  • Date: 2011
  • Type: Journal article
  • Relation: Journal of Global Optimization Vol. 51, no. 4 (2011), p. 677-688
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
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  • Description: The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained. © 2011 Springer Science+Business Media, LLC.

Comments on : Stability in linear optimization and related topics. A personal tour

  • Authors: Kruger, Alexander
  • Date: 2012
  • Type: Text , Journal article
  • Relation: TOP Vol. 20, no. 2 (2012), p. 255-257
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  • Description: The article presents a report on a wonderful tour in the area of stability analysis of linear (and not only linear) optimization undertaken in the last 15 years by the author and his team of collaborators. 15 years is a very short period for developing a mathematical theory. Nevertheless the scope of achievement presented in the article and the level of development of the theory are really impressive. The tour is full of attractions and the route is very carefully marked. Now the tour is on offer, and the author is eager to share its highlights with interested travelers.

Coverage in WLAN : Optimization model and algorithm

Coverage in WLAN with minimum number of access points

Dual sufficient characterizations of transversality properties

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Positivity Vol. 24, no. 5 (2020), p. 1313-1359
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe. © 2020, Springer Nature Switzerland AG.
  • Description: The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sk