Polytopes close to being simple
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 64, no. 1 (2020), p. 200-215
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that d-polytopes with at most d- 2 nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and d- 2 , showing that certain polytopes with more than two nonsimple vertices are reconstructible from their graphs. In particular, we prove that reconstructibility from graphs also holds for d-polytopes with d+ k vertices and at most d- k+ 3 nonsimple vertices, provided k
Refining the partition for multifold conic optimization problems
- Authors: Ramirez, Hector , Roshchina, Vera
- Date: 2020
- Type: Text , Journal article
- Relation: Optimization Vol. 69, no. 11 (2020), p. 2489-2507
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- Description: In this paper, we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in Bonnans and Ramirez [Perturbation analysis of second-order cone programming problems, Math Program. 2005;104(2-30):205-227] for conic optimization problems, and in Pena and Roshchina [A complementarity partition theorem for multifold conic systems, Math Program. 2013;142(1-2):579-589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular, that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions.
- Description: This research was partially supported by ANID (Chile) under REDES project number 180032 and by the Australian Research Council grant DE150100240. The second author was supported by FONDECYT (Fondo de Fomento al Desarrollo Cientifico y Tecnologico) regular projects 1160204 and 1201982, and Basal Program CMM-AFB 170001 (Comision Nacional de Investigacion Cientifica y Tecnologica), all from ANID (Chile).
Tennis influencers : the player effect on social media engagement and demand for tournament attendance
- Authors: Chmait, Nader , Westerbeek, Hans , Eime, Rochelle , Robertson, Sam , Sellitto, Carmine , Reid, Machar
- Date: 2020
- Type: Text , Journal article
- Relation: Telematics and Informatics Vol. 50, no. (2020), p.
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- Description: Understanding the interest of sports fans in professional tennis has valuable operational and marketing implications for tournament organisers, marketeers, player sponsors and the media. In sports, professional tennis in particular, the player effect on social media user engagement is still elusive. Using data from the 2019 Australian Open grand slam period, the authors examine Adler's (1985) theoretical construct in the context of sports and social media. A social listening tool is used to probe more than 2 million posts and comments mentioning elite male and female tennis players on four major social media channels: Twitter, Facebook, Instagram and YouTube, over the grand slam period. It is shown that the effect of professional tennis players on social media user engagement extends beyond their talent. A selection of players had a strong positive impact on prompting social media activity, even after accounting for factors related to their performance, the tournament rounds in which they were featured and the opponents against whom they played. Furthermore, the connection between social media research and sports economics is considered by examining the relationship between a player's effects on social media engagement and her/his differential influence on demand for tickets at the Australian Tennis Open. The authors further discuss how the social media star influence can be used, in combination with other quantitative measures, to optimise tennis tournament scheduling, determine player appearance fees and lift participation in the sport. © 2020 Elsevier Ltd
The effects of the no-touch gap on the no-touch bipolar radiofrequency ablation treatment of liver cancer : a numerical study using a two compartment model
- Authors: Yap, Shelley , Cheong, Jason , Foo, Ji , Ooi, Ean Tat , Ooi, Ean Hin
- Date: 2020
- Type: Text , Journal article
- Relation: Applied Mathematical Modelling Vol. 78, no. (2020), p. 134-147
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- Description: The no-touch bipolar radiofrequency ablation (RFA) for cancer treatment is advantageous primarily because of its capability to prevent tumour track seeding (TTS). In this technique, the RF probes are placed at a distance (no-touch gap) away from the tumour boundary. Ideally, the RF probes should be placed sufficiently far from the tumour in order to avoid TTS. However, having a gap that is too large can lead to ineffective ablation. This paper investigates how the selection of the no-touch gap can affect the tissue electrical and thermal responses during the no-touch bipolar RFA treatment. Simulations were carried out on a two compartment model using the finite element method. Results obtained indicated that a gap that is too large may lead to incomplete ablation and failure to achieve significant ablation margin. However, keeping the gap to be too small may not be clinically practical. It was suggested that the incomplete ablation and the insufficient ablation margin observed in some of the cases may require the placement of additional probes around the tumour. The present study stresses on the importance of identifying the optimal no-touch gap that can avoid TTS without compromising the treatment outcome. © 2019 Elsevier Inc.
A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections
- Authors: Cominetti, Roberto , Roshchina, Vera , Williamson, Andrew
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Vol. 68, no. 1 (2019), p. 3-12
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- Description: The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the ε-under-relaxed cyclic projection method converge when ε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in R3 for which the ε-under-relaxed cycles do not converge. © 2018 Informa UK Limited, trading as Taylor & Francis Group.
A difference of convex optimization algorithm for piecewise linear regression
- Authors: Bagirov, Adil , Taheri, Sona , Asadi, Soodabeh
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Industrial and Management Optimization Vol. 15, no. 2 (2019), p. 909-932
- Relation: http://purl.org/au-research/grants/arc/DP140103213
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- Description: The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed based on these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.
A sharp augmented Lagrangian-based method in constrained non-convex optimization
- Authors: Bagirov, Adil , Ozturk, Gurkan , Kasimbeyli, Refail
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 34, no. 3 (2019), p. 462-488
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- Description: In this paper, a novel sharp Augmented Lagrangian-based global optimization method is developed for solving constrained non-convex optimization problems. The algorithm consists of outer and inner loops. At each inner iteration, the discrete gradient method is applied to minimize the sharp augmented Lagrangian function. Depending on the solution found the algorithm stops or updates the dual variables in the inner loop, or updates the upper or lower bounds by going to the outer loop. The convergence results for the proposed method are presented. The performance of the method is demonstrated using a wide range of nonlinear smooth and non-smooth constrained optimization test problems from the literature.
Calmness of partially perturbed linear systems with an application to the central path
- Authors: Cánovas, Maria , Hall, Julian , López, Marco , Parra, Juan
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Vol. 68, no. 2-3 (2019), p. 465-483
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- Description: In this paper we develop point-based formulas for the calmness modulus of the feasible set mapping in the context of linear inequality systems with a fixed abstract constraint and (partially) perturbed linear constraints. The case of totally perturbed linear systems was previously analyzed in [Canovas MJ, Lopez MA, Parra J, et al. Calmness of the feasible set mapping for linear inequality systems. Set-Valued Var Anal. 2014;22:375-389, Section 5]. We point out that the presence of such an abstract constraint yields the current paper to appeal to a notable different methodology with respect to previous works on the calmness modulus in linear programming. The interest of this model comes from the fact that partially perturbed systems naturally appear in many applications. As an illustration, the paper includes an example related to the classical central path construction. In this example we consider a certain feasible set mapping whose calmness modulus provides a measure of the convergence of the central path. Finally, we underline the fact that the expression for the calmness modulus obtained in this paper is (conceptually) implementable as far as it only involves the nominal data.
Characterizations of nonsmooth robustly quasiconvex functions
- Authors: Bui, Hoa , Khanh, Pham , Tran, Thi
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 180, no. 3 (2019), p. 775-786
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- Description: Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fréchet subdifferentials in Asplund spaces. The first criterion extends to such spaces a result established by Barron et al. (Discrete Contin Dyn Syst Ser B 17:1693–1706, 2012). The second criterion is totally new even if it is applied to lower semicontinuous functions on finite-dimensional spaces. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Convexity and closedness in stable robust duality
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 13, no. 2 (2019), p. 325-339
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.
Extremality, stationarity and generalized separation of collections of sets
- Authors: Bui, Hoa , Kruger, Alexander
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 182, no. 1 (2019), p. 211-264
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- Description: The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analysed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining ‘extremal’ statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.
New Farkas-type results for vector-valued functions : A non-abstract approach
- Authors: Dinh, Nguyen , Goberna, Miguel , Long, Dang , Lopez-Cerda, Marco
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 182, no. 1 (2019), p. 4-29
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- Description: This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357-390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.
On the reconstruction of polytopes
- Authors: Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2019
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302. http://purl.org/au-research/grants/arc/DP180100602
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- Description: Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Outer limits of subdifferentials for min–max type functions
- Authors: Eberhard, Andrew , Roshchina, Vera , Sang, Tian
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Vol. 68, no. 7 (2019), p. 1391-1409
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- Description: We generalize the outer subdifferential construction suggested by Cánovas, Henrion, López and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential posed by Li, Meng and Yang.
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. http://purl.org/au-research/grants/arc/DP180100602
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- 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.
Two curve Chebyshev approximation and its application to signal clustering
- Authors: Sukhorukova, Nadezda
- Date: 2019
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 356, no. (2019), p. 42-49
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved. In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved.
Variational analysis Down Under open problem session
- Authors: Bui, Hoa , Lindstrom, Scott , Roshchina, Vera
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 182, no. 1 (2019), p. 430-437
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- Description: We state the problems discussed in the open problem session at Variational Analysis Down Under conference held in honour of Prof. Asen Dontchev on 19-21 February 2018 at Federation University Australia.
Double bundle method for finding clarke stationary points in nonsmooth dc programming
- Authors: Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko , Taheri, Sona
- Date: 2018
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 28, no. 2 (2018), p. 1892-1919
- Relation: http://purl.org/au-research/grants/arc/DP140103213
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- Description: The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.
Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations
- Authors: Gaudioso, Manlio , Giallombardo, Giovanni , Miglionico, Giovanna , Bagirov, Adil
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 71, no. 1 (2018), p. 37-55
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- Description: We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems. © 2017, Springer Science+Business Media, LLC.
Nonsmooth DC programming approach to clusterwise linear regression : Optimality conditions and algorithms
- Authors: Bagirov, Adil , Ugon, Julien
- Date: 2018
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 33, no. 1 (2018), p. 194-219
- Relation: http://purl.org/au-research/grants/arc/DP140103213
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- Description: The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets. © 2017 Informa UK Limited, trading as Taylor & Francis Group.