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Journal ArticleDOI

On solving a non-convex quadratic programming problem involving resistance distances in graphs

01 Apr 2020-Annals of Operations Research (Springer US)-Vol. 287, Iss: 2, pp 643-651
TL;DR: This paper considers the question of solving the quadratic programming problem of finding maximum of x T R x subject to x being a nonnegative vector with sum 1 and shows that for the class of simple graphs with resistance distance matrix ( R ) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratics programming problem.
Abstract: Quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12–20, 2012), Bapat and Neogy (Ann Oper Res 243:365–373, 2016). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of $$x^{T}Rx$$ subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix (R) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented.
Citations
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Posted Content
TL;DR: A theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes, and shows how the maximum-variance distribution can be interpreted as a core-periphery measure.
Abstract: We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.

12 citations


Cites background from "On solving a non-convex quadratic p..."

  • ...This result was shown before in [6] in context of calculating the average weighted resistance distance on a graph....

    [...]

  • ...[6] D....

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  • ...These results are generalized from the geodesic graph distance to the effective resistance in [5, 6], where it is shown that the corresponding maximum variance problem can be solved efficiently....

    [...]

Journal ArticleDOI
TL;DR: In this article , the authors developed a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes.
Abstract: We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and we characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.

4 citations

Journal ArticleDOI
TL;DR: In this paper , a nonconvex function activated implicit ZNN (NIZNN) model and a noise-suppressing implicit zeroing neural network (NNSIZNN) were proposed by the inspiration of the conventional ZNN model from a control-based framework, which guarantees an effective solution for TVQPPs in the presence of different noises.
Abstract: Time-varying quadratic programming problems (TVQPPs) subject to equality and inequality constraints with different measurement noises often arise in the fields of scientific and economic research. Noises are always ubiquitous and unavoidable in practical problems. However, most existing methods usually assume that the computing process is free of measurement noise or the denoising has been monitored before the calculation. The zeroing neural network (ZNN), a significant neurodynamic approach, has presented potent abilities to compute a great variety of time-varying zeroing problems with odd monotonically increasing activation functions, but the existing results on ZNN cannot deal with the inequality constraints in the TVQPPs. In this article, a nonconvex function activated implicit ZNN (NIZNN) model and a noise-suppressing implicit ZNN (NNSIZNN) model, which is also known as the ZNN-based model, are proposed by the inspiration of the conventional ZNN model from a control-based framework. It guarantees an effective solution for TVQPPs in the presence of different noises. The ZNN-based model allows nonconvex sets for projection operation in activation functions and incorporates noise-tolerant techniques for handling different noises arising in TVQPPs. Besides, theoretical analyses show that the ZNN-based model globally converges to the time-varying optimization solution of TVQPPs in the presence of noises. In addition, an illustrative example is provided to substantiate the efficiency and robustness of the ZNN-based model for online solving TVQPPs with inherent tolerance to noises. Moreover, the ZNN-based model is applied to the economic model, which provides and investigates their computational efficiency and superiority. Finally, an application example to repetitive motion generation of four-wheel omnidirectional mobile robot is simulated to substantiate the feasibility and superiority of the developed NNSIZNN model for online solving TVQPPs with nonconvex constraints and measurement noise.

1 citations

Book ChapterDOI
02 Feb 2020
TL;DR: In this article, the diversity of copositive optimization formulations in different domains of optimization is demonstrated, and the role of copositivity for local and global optimality conditions is discussed.
Abstract: Recently, copositive optimization has received a lot of attention to the Operational Research community, and it is rapidly expanding and becoming a fertile field of research. In this chapter, we demonstrate the diversity of copositive formulations in different domains of optimization: continuous, discrete, and stochastic optimization problems. Further, we discuss the role of copositivity for local and global optimality conditions. Finally, we talk about some applications of copositive optimization in graph theory and game theory.
References
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Book
18 Feb 1992
TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.
Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

2,897 citations

Journal ArticleDOI
TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.
Abstract: Some simple constructive proofs are given of solutions to the matric system Mz − ω = q; z ≧ 0; ω ≧ 0; and zT ω = 0, for various kinds of data M, q, which embrace the quadratic programming problem and the problem of finding equilibrium points of bimatrix games. The general scheme is, assuming non-degeneracy, to generate an adjacent extreme point path leading to a solution. The scheme does not require that some functional be reduced.

966 citations

Book
17 Dec 2011
TL;DR: The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.
Abstract: This book deals with the analysis of the structure of complex networks by combining results from graph theory, physics, and pattern recognition. The book is divided into two parts. 11 chapters are dedicated to the development of theoretical tools for the structural analysis of networks, and 7 chapters are illustrating, in a critical way, applications of these tools to real-world scenarios. The first chapters provide detailed coverage of adjacency and metric and topological properties of networks, followed by chapters devoted to the analysis of individual fragments and fragment-based global invariants in complex networks. Chapters that analyse the concepts of communicability, centrality, bipartivity, expansibility and communities in networks follow. The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.

585 citations

Journal ArticleDOI
Éva Tardos1
TL;DR: This work gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities.
Abstract: Khachiyan, and recently Karmarkar, gave polynomial algorithms to solve the linear programming problem. These algorithms have a small theoretical drawback; namely, the number of arithmetic steps depends on the size of the input numbers. We present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix, but is independent of the size of the numbers in the right-hand side and objective vectors. In particular, it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities. The algorithm makes use of an existing polynomial linear programming algorithm. The problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open.

495 citations

Book
01 Jan 2014
TL;DR: In this article, the authors present a matrix game based on graph games, where the objective is to find the positive definite completion problem in a graph. But the game is not suitable for children.
Abstract: Preliminaries.- Incidence Matrix.- Adjacency Matrix.- Laplacian Matrix.- Cycles and Cuts.- Regular Graphs.- Line Graph of a Tree.- Algebraic Connectivity.- Distance Matrix of a Tree.- Resistance Distance.- Laplacian Eigenvalues of Threshold Graphs.- Positive Definite Completion Problem.- Matrix Games Based on Graphs.

482 citations