A new proof of Balinski's theorem on the connectivity of polytopes
- Authors: Pineda-Villavicencio, Guillermo
- Date: 2021
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 344, no. 7 (2021), p.
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- Description: Balinski (1961) proved that the graph of a d-dimensional convex polytope is d-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly d vertices; some of which appear to be new. © 2021 Elsevier B.V.
Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric
- Authors: Beer, Gerald , Cánovas, M. J. , López, Marco , Parra, Juan
- Date: 2021
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 189, no. 1-2 (2021), p. 75-98. https://purl.org/au-research/grants/arc/DP180100602
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Correction: The article “Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric”, written by Beer,G., Cánovas, M.J., López, M.A., Parra, J.was originally published Online First without Open Access. After publication in volume 189, issue 1–2, page 75–98 the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © The Author(s) 2020 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License. https://doi.org/10.1007/s10107-021-01751-x
- Description: This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.
Assessing cohesion of the rocks proposing a new intelligent technique namely group method of data handling
- Authors: Chen, Wusi , Khandelwal, Manoj , Murlidhar, Bhatawdekar , Bui, Dieu , Tahir, Mahmood , Katebi, Javad
- Date: 2020
- Type: Text , Journal article
- Relation: Engineering with Computers Vol. 36, no. 2 (2020), p. 783-793
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- Description: In this study, evaluation and prediction of rock cohesion is assessed using multiple regression as well as group method of data handling (GMDH). It is a well-known fact that cohesion is the most crucial rock shear strength parameter, which is a key parameter for the stability evaluation of some geotechnical structures such as rock slope. To fulfill the aim of this study, a database of three model input parameters, i.e., p wave velocity, uniaxial compressive strength and Brazilian tensile strength and one model output, which is cohesion of limestone samples was prepared and utilized by GMDH. Different GMDH models with neurons and layers and selection pressure were tested and assessed. It was found that GMDH model number 4 (with 8 layers) shows the best performance among all of tested models between the input and output parameters for the prediction and assessment of rock cohesion with coefficient of determination (R2) values of 0.928 and 0.929, root mean square error values of 0.3545 and 0.3154 for training and testing datasets, respectively. Multiple regression analysis was also performed on the same database and R2 values were obtained as 0.8173 and 0.8313 between input and output parameters for the training and testing of the models, respectively. The GMDH technique developed in this study is introduced as a new model in field of rock shear strength parameters. © 2019, Springer-Verlag London Ltd., part of Springer Nature.
Effects of a proper feature selection on prediction and optimization of drilling rate using intelligent techniques
- Authors: Liao, Xiufeng , Khandelwal, Manoj , Yang, Haiqing , Koopialipoor, Mohammadreza , Murlidhar, Bhatawdekar
- Date: 2020
- Type: Text , Journal article
- Relation: Engineering with Computers Vol. 36, no. 2 (Apr 2020), p. 499-510
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- Description: One of the important factors during drilling times is the rate of penetration (ROP), which is controlled based on different variables. Factors affecting different drillings are of paramount importance. In the current research, an attempt was made to better recognize drilling parameters and optimize them based on an optimization algorithm. For this purpose, 618 data sets, including RPM, flushing media, and compressive strength parameters, were measured and collected. After an initial investigation, the compressive strength feature of samples, which is an important parameter from the rocks, was used as a proper criterion for classification. Then using intelligent systems, three different levels of the rock strength and all data were modeled. The results showed that systems which were classified based on compressive strength showed a better performance for ROP assessment due to the proximity of features. Therefore, these three levels were used for classification. A new artificial bee colony algorithm was used to solve this problem. Optimizations were applied to the selected models under different optimization conditions, and optimal states were determined. As determining drilling machine parameters is important, these parameters were determined based on optimal conditions. The obtained results showed that this intelligent system can well improve drilling conditions and increase the ROP value for three strength levels of the rocks. This modeling system can be used in different drilling operations.
On computation of optimal strategies in oligopolistic markets respecting the cost of change
- Authors: Outrata, Jiri , Valdman, Jan
- Date: 2020
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 92, no. 3 (2020), p. 489-509
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with a class of parameterized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability, under appropriate conditions, is established. This theory is then applied to evolution of an oligopolistic market in which the firms adapt their production strategies to changing input costs, while each change of the production is associated with some “costs of change”. We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
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.
Perturbation of error bounds
- Authors: Kruger, Alexander , López, Marco , Théra, Michel
- Date: 2018
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.
Set regularities and feasibility problems
- Authors: Kruger, Alexander , Luke, Russell , Thao, Nguyen
- Date: 2018
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 168, no. 1-2 (2018), p. 279-311
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms
On the Aubin property of a class of parameterized variational systems
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2017
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 86, no. 3 (2017), p. 443-467
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with a new sharp condition ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class encompasses various types of parameterized variational inequalities/generalized equations with fairly general constraint sets. The new condition requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested procedure is illustrated by means of examples. © 2017, Springer-Verlag GmbH Germany.
Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions
- Authors: Chen, Yi , Gao, David
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 64, no. 3 (2016), p. 417-431
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- Description: This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of 4th-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical duality–triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples. © 2014, Springer Science+Business Media New York.
Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain
- Authors: Adly, Samir , Hantoute, Abderrahim , Thera, Michel
- Date: 2016
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 157, no. 2 (2016), p. 349-374
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- Description: The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in the previous paper (Adly et al. in Nonlinear Anal 75(3): 985–1008, 2012). This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty; this includes in a natural way the finite-dimensional case. The current setting leads to simplified, more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials. Some consequences of the viability of closed sets are given. Our analysis makes use of standard tools from convex and variational analysis. © 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.
Directional metric regularity of multifunctions
- 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
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- 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.
Gradient-free method for nonsmooth distributed optimization
- Authors: Li, Jueyou , Wu, Changzhi , Wu, Zhiyou , Long, Qiang
- Date: 2014
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol.61, no.2 (March 2014), p.325-340
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- Description: In this paper, we consider a distributed nonsmooth optimization problem over a computational multi-agent network. We first extend the (centralized) Nesterov’s random gradient-free algorithm and Gaussian smoothing technique to the distributed case. Then, the convergence of the algorithm is proved. Furthermore, an explicit convergence rate is given in terms of the network size and topology. Our proposed method is free of gradient, which may be preferred by practical engineers. Since only the cost function value is required, our method may suffer a factor up to d (the dimension of the agent) in convergence rate over that of the distributed subgradient-based methods in theory. However, our numerical simulations show that for some nonsmooth problems, our method can even achieve better performance than that of subgradient-based methods, which may be caused by the slow convergence in the presence of subgradient.
New largest known graphs of diameter 6
- 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
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- Description: In the pursuit of obtaining largest graphs of given maximum degree
- Description: 2003007890