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.
Set regularities and feasibility problems
- Kruger, Alexander, Luke, Russell, Thao, Nguyen
- 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
- Full Text:
- Reviewed:
- 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
- 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
- Full Text:
- Reviewed:
- 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
Levels of explainable artificial intelligence for human-aligned conversational explanations
- Dazeley, Richard, Vamplew, Peter, Foale, Cameron, Young, Cameron, Aryal, Sunil, Cruz, Francisco
- Authors: Dazeley, Richard , Vamplew, Peter , Foale, Cameron , Young, Cameron , Aryal, Sunil , Cruz, Francisco
- Date: 2021
- Type: Text , Journal article
- Relation: Artificial Intelligence Vol. 299, no. (2021), p.
- Full Text:
- Reviewed:
- Description: Over the last few years there has been rapid research growth into eXplainable Artificial Intelligence (XAI) and the closely aligned Interpretable Machine Learning (IML). Drivers for this growth include recent legislative changes and increased investments by industry and governments, along with increased concern from the general public. People are affected by autonomous decisions every day and the public need to understand the decision-making process to accept the outcomes. However, the vast majority of the applications of XAI/IML are focused on providing low-level ‘narrow’ explanations of how an individual decision was reached based on a particular datum. While important, these explanations rarely provide insights into an agent's: beliefs and motivations; hypotheses of other (human, animal or AI) agents' intentions; interpretation of external cultural expectations; or, processes used to generate its own explanation. Yet all of these factors, we propose, are essential to providing the explanatory depth that people require to accept and trust the AI's decision-making. This paper aims to define levels of explanation and describe how they can be integrated to create a human-aligned conversational explanation system. In so doing, this paper will survey current approaches and discuss the integration of different technologies to achieve these levels with Broad eXplainable Artificial Intelligence (Broad-XAI), and thereby move towards high-level ‘strong’ explanations. © 2021 Elsevier B.V.
- Authors: Dazeley, Richard , Vamplew, Peter , Foale, Cameron , Young, Cameron , Aryal, Sunil , Cruz, Francisco
- Date: 2021
- Type: Text , Journal article
- Relation: Artificial Intelligence Vol. 299, no. (2021), p.
- Full Text:
- Reviewed:
- Description: Over the last few years there has been rapid research growth into eXplainable Artificial Intelligence (XAI) and the closely aligned Interpretable Machine Learning (IML). Drivers for this growth include recent legislative changes and increased investments by industry and governments, along with increased concern from the general public. People are affected by autonomous decisions every day and the public need to understand the decision-making process to accept the outcomes. However, the vast majority of the applications of XAI/IML are focused on providing low-level ‘narrow’ explanations of how an individual decision was reached based on a particular datum. While important, these explanations rarely provide insights into an agent's: beliefs and motivations; hypotheses of other (human, animal or AI) agents' intentions; interpretation of external cultural expectations; or, processes used to generate its own explanation. Yet all of these factors, we propose, are essential to providing the explanatory depth that people require to accept and trust the AI's decision-making. This paper aims to define levels of explanation and describe how they can be integrated to create a human-aligned conversational explanation system. In so doing, this paper will survey current approaches and discuss the integration of different technologies to achieve these levels with Broad eXplainable Artificial Intelligence (Broad-XAI), and thereby move towards high-level ‘strong’ explanations. © 2021 Elsevier B.V.
A new proof of Balinski's theorem on the connectivity of polytopes
- Pineda-Villavicencio, Guillermo
- Authors: Pineda-Villavicencio, Guillermo
- Date: 2021
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 344, no. 7 (2021), p.
- Full Text:
- Reviewed:
- 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.
- Authors: Pineda-Villavicencio, Guillermo
- Date: 2021
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 344, no. 7 (2021), p.
- Full Text:
- Reviewed:
- 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.
Assessing cohesion of the rocks proposing a new intelligent technique namely group method of data handling
- Chen, Wusi, Khandelwal, Manoj, Murlidhar, Bhatawdekar, Bui, Dieu, Tahir, Mahmood, Katebi, Javad
- 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
- Full Text:
- Reviewed:
- 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.
- 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
- Full Text:
- Reviewed:
- 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
- Liao, Xiufeng, Khandelwal, Manoj, Yang, Haiqing, Koopialipoor, Mohammadreza, Murlidhar, Bhatawdekar
- 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
- Full Text:
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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.
- 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
- Full Text:
- Reviewed:
- 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.
- 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
- Full Text:
- Reviewed:
- 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.
Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain
- Adly, Samir, Hantoute, Abderrahim, Thera, Michel
- 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.
- Authors: Adly, Samir , Hantoute, Abderrahim , Thera, Michel
- Date: 2016
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 157, no. 2 (2016), p. 349-374
- Full Text:
- Reviewed:
- 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.
The linkedness of cubical polytopes: the cube
- Bui, Hoa, Pineda-Villavicencio, Guillermo, Ugon, Julien
- Authors: Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien
- Date: 2021
- Type: Text , Journal article
- Relation: Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph G is strongly k-linked if it has at least 2k + 1 vertices and, for every vertex v of G, the subgraph G − v is k-linked. We show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly
- Authors: Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien
- Date: 2021
- Type: Text , Journal article
- Relation: Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph G is strongly k-linked if it has at least 2k + 1 vertices and, for every vertex v of G, the subgraph G − v is k-linked. We show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly
Syntactic characterizations of polynomial time optimization classes
- Authors: Manyem, Prabhu
- Date: 2008
- Type: Journal article
- Relation: Chicago Journal of Theoretical Computer Science Vol. 2008, no. (2008), p.
- Full Text:
- Reviewed:
- Description: The characterization of important complexity classes by logical descriptions has been an important and prolific area of Descriptive complexity. However, the central focus of the research has been the study of classes like P, NP, L and NL, corresponding to decision problems (e.g. the characterization of NP by Fagin [Fag74] and of P by Gradel [E. 91]). In contrast, optimization problems have received much less attention. Optimization problems corresponding to the NP class have been characterized in terms of logic expressions by Papadimitriou and Yannakakis, Panconesi and Ranjan, Kolaitis and Thakur, Khanna et al, and by Zimand. In this paper, we attempt to characterize the optimization versions of P via expressions in second order logic, many of them using universal Horn formulae with successor relations. These results nicely complement those of Kolaitis and Thakur [KT94] for polynomially bounded NP-optimization problems. The polynomially bounded versions of maximization and minimization problems are treated first, and then the maximization problems in the not necessarily polynomially bounded class.
- Description: 2003006662
- Authors: Manyem, Prabhu
- Date: 2008
- Type: Journal article
- Relation: Chicago Journal of Theoretical Computer Science Vol. 2008, no. (2008), p.
- Full Text:
- Reviewed:
- Description: The characterization of important complexity classes by logical descriptions has been an important and prolific area of Descriptive complexity. However, the central focus of the research has been the study of classes like P, NP, L and NL, corresponding to decision problems (e.g. the characterization of NP by Fagin [Fag74] and of P by Gradel [E. 91]). In contrast, optimization problems have received much less attention. Optimization problems corresponding to the NP class have been characterized in terms of logic expressions by Papadimitriou and Yannakakis, Panconesi and Ranjan, Kolaitis and Thakur, Khanna et al, and by Zimand. In this paper, we attempt to characterize the optimization versions of P via expressions in second order logic, many of them using universal Horn formulae with successor relations. These results nicely complement those of Kolaitis and Thakur [KT94] for polynomially bounded NP-optimization problems. The polynomially bounded versions of maximization and minimization problems are treated first, and then the maximization problems in the not necessarily polynomially bounded class.
- Description: 2003006662
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
- Full Text:
- Reviewed:
- 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.
- 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
- Full Text:
- Reviewed:
- 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.
Polytopes close to being simple
- Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David
- 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
- Full Text:
- Reviewed:
- 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
- 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
- Full Text:
- Reviewed:
- 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
Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric
- Beer, Gerald, Cánovas, M. J., López, Marco, Parra, Juan
- 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
- Full Text:
- Reviewed:
- 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.
- 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
- Full Text:
- Reviewed:
- 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.