Efficient deterministic algorithm for huge-sized noisy sensor localization problems via canonical duality theory
- Authors: Latorre, Vittorio , Gao, David
- Date: 2021
- Type: Text , Journal article
- Relation: IEEE Transactions on Cybernetics Vol. 51, no. 10 (2021), p. 5069-5081
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- Description: This paper presents a new deterministic method and a polynomial-time algorithm for solving general huge-sized sensor network localization problems. The problem is first formulated as a nonconvex minimization, which was considered as an NP-hard based on conventional theories. However, by the canonical duality theory, this challenging problem can be equivalently converted into a convex dual problem. By introducing a new optimality measure, a powerful canonical primal-dual interior (CPDI) point algorithm is developed which can solve efficiently huge-sized problems with hundreds of thousands of sensors. The new method is compared with the popular methods in the literature. Results show that the CPDI algorithm is not only faster than the benchmarks but also much more accurate on networks affected by noise on the distances. © 2013 IEEE.
Generalized bregman envelopes and proximity operators
- Authors: Burachik, Regina , Dao, Minh , Lindstrom, Scott
- Date: 2021
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 190, no. 3 (2021), p. 744-778
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- Description: Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Incremental DC optimization algorithm for large-scale clusterwise linear regression
- Authors: Bagirov, Adil , Taheri, Sona , Cimen, Emre
- Date: 2021
- Type: Text , Journal article
- Relation: Journal of Computational and Applied Mathematics Vol. 389, no. (2021), p. 1-17
- Relation: https://purl.org/au-research/grants/arc/DP190100580
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- Description: The objective function in the nonsmooth optimization model of the clusterwise linear regression (CLR) problem with the squared regression error is represented as a difference of two convex functions. Then using the difference of convex algorithm (DCA) approach the CLR problem is replaced by the sequence of smooth unconstrained optimization subproblems. A new algorithm based on the DCA and the incremental approach is designed to solve the CLR problem. We apply the Quasi-Newton method to solve the subproblems. The proposed algorithm is evaluated using several synthetic and real-world data sets for regression and compared with other algorithms for CLR. Results demonstrate that the DCA based algorithm is efficient for solving CLR problems with the large number of data points and in particular, outperforms other algorithms when the number of input variables is small. © 2020 Elsevier B.V.
An existence result for quasi-equilibrium problems via Ekeland’s variational principle
- Authors: Cotrina, John , Théra, Michel , Zúñiga, Javier
- Date: 2020
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 187, no. 2 (2020), p. 336-355
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- Description: This paper deals with the existence of solutions to equilibrium and quasi-equilibrium problems without any convexity assumption. Coverage includes some equivalences to the Ekeland variational principle for bifunctions and basic facts about transfer lower continuity. An application is given to systems of quasi-equilibrium problems. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
- Description: Research of M. Théra is supported by the Australian Research Council (ARC) Grant DP160100854 and benefited from the support of the FMJH Program PGMO and from the support of EDF. http://purl.org/au-research/grants/arc/DP160100854
Orthogonality in locally convex spaces : two nonlinear generalizations of Neumann's lemma
- Authors: Barbagallo, Annamaria , Ernst, Octavian-Emil , Théra, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 484, no. 1 (Apr 2020), p. 18
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- Description: In this note we prove a symmetric version of the Neumann lemma as well as a symmetric version of the Soderlind-Campanato lemma. We establish in this way two partial generalizations of the well-known Casazza-Christenses lemma. This work is related to the Birkhoff-James orthogonality and to the concept of near operators introduced by S. Campanato. (C) 2019 Published by Elsevier Inc.
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.
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 semiregularity of mappings
- Authors: Cibulka, Radek , Fabian, Marian , Kruger, Alexander
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.
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.
A comparison of bidding strategies for online auctions using fuzzy reasoning and negotiation decision functions
- Authors: Kaur, Preetinder , Goyal, Madhu , Lu, Jie
- Date: 2017
- Type: Text , Journal article
- Relation: IEEE Transactions on Fuzzy Systems Vol. 25, no. 2 (2017), p. 425-438
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- Description: Bidders often feel challenged when looking for the best bidding strategies to excel in the competitive environment of multiple and simultaneous online auctions for same or similar items. Bidders face complicated issues for deciding which auction to participate in, whether to bid early or late, and how much to bid. In this paper, we present the design of bidding strategies, which aim to forecast the bid amounts for buyers at a particular moment in time based on their bidding behavior and their valuation of an auctioned item. The agent develops a comprehensive methodology for final price estimation, which designs bidding strategies to address buyers' different bidding behaviors using two approaches: Mamdani method with regression analysis and negotiation decision functions. The experimental results show that the agents who follow fuzzy reasoning with a regression approach outperform other existing agents in most settings in terms of their success rate and expected utility.
A unifying approach to robust convex infinite optimization duality
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.
Characterization theorem for best polynomial spline approximation with free knots, variable degree and fixed tails
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964
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- Description: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.
Farkas-type results for vector-valued functions with applications
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390
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- Description: The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.
Applying UGF Concept to Enhance the Assessment Capability of FMEA
- Authors: Khorshidi, Hadi , Gunawan, Indra , Ibrahim, Yousef
- Date: 2016
- Type: Text , Journal article
- Relation: Quality and Reliability Engineering International Vol. 32, no. 3 (Apr 2016), p. 1085-1093
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- Description: The purpose of this paper is to propose a modified version of Failure Mode and Effects Analysis (FMEA) to alleviate its drawbacks. FMEA is an important tool in risk evaluation and finding the priority of potential failure modes for corrective actions. In the proposed method, the Universal Generating Function (UGF) approach has been used to improve the assessment capability of the conventional Risk Priority Number (RPN) in ranking. The new method is named as URPN. It generates the most number of unique values in comparison with the previous methods and considers relative importance for the parameters while it is easy to compute. More unique numbers help to avoid from having the same priority level for different failure modes which represent various risk levels. A case study has been employed to demonstrate that the URPN not only can improve the shortcomings but also is able to provide accurate values for risk assessment. Copyright (c) 2015 John Wiley & Sons, Ltd.
Borwein-Preiss variational principle revisited
- 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
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- 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.
Chebyshev approximation by linear combinations of fixed knot polynomial splines with weighting functions
- Authors: Sukhorukova, Nadezda , Ugon, Julien
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 2 (2016), p. 536-549
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- Description: In this paper, we derive conditions for best uniform approximation by fixed knots polynomial splines with weighting functions. The theory of Chebyshev approximation for fixed knots polynomial functions is very elegant and complete. Necessary and sufficient optimality conditions have been developed leading to efficient algorithms for constructing optimal spline approximations. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). In this paper, we extend these results to the case when the model function is a product of fixed knots polynomial splines (whose parameters are subject to optimization) and other functions (whose parameters are predefined). This problem is nonsmooth, and therefore, we make use of convex and nonsmooth analysis to solve it.
Directional Holder metric regularity
- 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
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- 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.
Nonlinear metric subregularity
- Authors: Kruger, Alexander
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 171, no. 3 (2016), p. 820-855
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this article, we investigate nonlinear metric subregularity properties of set-valued mappings between general metric or Banach spaces. We demonstrate that these properties can be treated in the framework of the theory of (linear) error bounds for extended real-valued functions of two variables developed in Kruger (Error bounds and metric subregularity. Optimization 64(1):49-79, 2015). Several primal and dual space local quantitative and qualitative criteria of nonlinear metric subregularity are formulated. The relationships between the criteria are established and illustrated.
A BMI approach to guaranteed cost control of discrete-time uncertain system with both state and input delays
- Authors: Zhou, Xiaojun , Dong, Tianxue , Tang, Xiaolin , Yang, Chunhua , Gui, Weihua
- Date: 2015
- Type: Text , Journal article
- Relation: Optimal Control Applications and Methods Vol. 36, no. 6 (2015), p. 844-852
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- Description: In this study, the guaranteed cost control of discrete time uncertain system with both state and input delays is considered. Sufficient conditions for the existence of a memoryless state feedback guaranteed cost control law are given in the bilinear matrix inequality form, which needs much less auxiliary matrix variables and storage space. Furthermore, the design of guaranteed cost controller is reformulated as an optimization problem with a linear objective function, bilinear, and linear matrix inequalities constraints. A nonlinear semi-definite optimization solver - PENLAB is used as a solution technique. A numerical example is given to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.