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Generalized Unitary Joint Diagonalization Algorithm Based on Approximate Givens Rotations

IEEE SIGNAL PROCESSING LETTERS 2024年 31卷 101-105页

作者： Azzouz, Malika Mesloub, Ammar Abed-Meraim, Karim Belouchrani, Adel Ecole Mil Polytech Lab Traitement Signal Algiers 16111 Algeria Univ Orleans F-45100 Orleans France Ecole Natl Polytech Algiers 16200 Algeria

Like the Joint Diagonalization of a unique set of matrices, Generalized Joint Diagonalization is an algebraic problem encountered in different applications such as data fusion and blind source separation. This letter proposes a new generalized unitary joint diagonalization approach based on the Jacobi iterative scheme using Givens rotations and simplified criterion, introducing three approximations. These approximations allowed us to reach a simultaneous estimation of different parameters. The first appears in the simplified criterion composed of entries doubly affected by the Givens rotations. The second approximation is in the Givens parameter using a small amplitude angle. The last approximation resides in keeping only the first order of transformed entries. Numerical experiments, including examples of joint blind audio source separation, are provided. The results show the effectiveness of the developed algorithm as compared to existing ones. The simultaneous estimation of different Givens rotations improves the algorithm's computation complexity and convergence rate.

关键词： Jacobian matrices Mathematical models Covariance matrices approximation algorithms Convergence Signal processing algorithms Estimation Generalized unitary joint diagonalization (GUJD) Givens rotations Jacobi iterative algorithm joint blind source separation (JBSS)

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One-Stage O(N log N) Algorithm for Generating Nested Rank-Minimized Representation of Electrically Large Volume Integral Equations

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IEEE JOURNAL ON MULTISCALE AND MULTIPHYSICS COMPUTATIONAL TECHNIQUES 2025年 10卷 169-178页

作者： Wang, Yifan Jiao, Dan Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA

In this paper, we develop a new one-stage O(N log N) algorithm to generate a rank-minimized H-2-representation of electrically large volume integral equations (VIEs), which significantly reduces the CPU run time of state-ofthe-art algorithms for completing the same task. Unlike existing two-stage algorithms, this new algorithm requires only one stage to build nested cluster bases. The cluster basis is obtained directly from the interaction between a cluster and its admissible clusters composed of real or auxiliary ones that cover all interaction directions. Furthermore, the row and column pivots of the resultant low- rank representation are chosen from the source and observer points in an analytical way without the need for numerically finding them. This further speeds up the computation. Numerical experiments on a suite of electrically large 3D scattering problems have demonstrated the efficiency and accuracy of the proposed new algorithm.

关键词： Accuracy approximation algorithms Complexity theory Clustering algorithms Integral equations Costs Indexes Training Three-dimensional displays Scattering Volume integral equations electrically large analysis fast solvers H-2 matrix compression rank-minimized compression nested representation low-rank representation

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Reinforcement Learning Solutions to Stochastic Multi-Agent Graphical Games With Multiplicative Noise

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2025年

作者： Zhang, Hanfang Wang, Bing-Chang Cao, Ying Shandong Univ Sch Control Sci & Engn Jinan 250061 Peoples R China

This paper investigates reinforcement learning algorithms for discrete-time stochastic multi-agent graphical games with multiplicative noise. The Bellman optimality equation for stochastic multi-agent graphical games is obtained by using the optimality principle. A Nash equilibrium can be reached when each agent executes a strategy in terms of Bellman optimality equation. To circumvent the difficulty of solving the coupled Bellman equation, a value iteration heuristic dynamic programming (HDP) algorithm is designed and its convergence is shown. To solve multi-agent graphical games online, the HDP algorithm based on the actor-critic framework is designed to approximate Nash equilibrium solutions. The effectiveness of the algorithm is verified by two numerical simulation examples.

关键词： Games Heuristic algorithms Mathematical models Noise Multi-agent systems approximation algorithms Reinforcement learning Nash equilibrium Vectors Optimal control Multi-agent graphical games Bellman optimality equation HDP algorithm

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Tight approximation bounds for the LPT rule applied to identical parallel machines with small jobs

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JOURNAL OF SCHEDULING 2022年第6期25卷 721-740页

作者： Lee, Myungho Lee, Kangbok Pinedo, Michael Postech 77 Cheongam Ro Pohang Si 37673 Gyeongsangbuk D South Korea NYU Stern Sch Business New York NY 10012 USA

We consider a scheduling problem with m identical machines in parallel and the minimum makespan objective. The Longest Processing Time first (LPT) rule is a well-known approximation algorithm for this problem. Although its worst-case approximation ratio has been determined theoretically, it is known that the worst-case approximation ratio of LPT can be smaller with instances of smaller processing times. We assume that each job's processing time is not longer than 1/k times the optimal makespan for a given integer k. We derive the worst-case approximation ratio of the LPT algorithm in terms of parameters k and m. For that purpose, we divide the whole set of instances of the original problem into classes defined by different values of parameters k and m. On each of those classes, we derive an exact upper bound on the worst-case performance ratio as a function of parameters k and m. We also show that there exist classes of instances for which our worst-case approximation ratio is better than previous bounds. Our bound can complement previous research in terms of the performance analysis of LPT.

关键词： Identical parallel machine scheduling Makespan minimization LPT rule approximation algorithms Processing time restriction

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User Association for Symbiotic Spectrum and Service Sharing Among Multiple Mobile Network Operators

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2024年第3期23卷 2167-2180页

作者： He, Shizhao Ge, Jungang Liang, Ying-Chang Univ Elect Sci & Technol China UESTC Natl Key Lab Wireless Commun Chengdu 611731 Peoples R China Univ Elect Sci & Technol China UESTC Ctr Intelligent Networking & Commun CINC Chengdu 611731 Peoples R China Univ Elect Sci & Technol China Ctr Intelligent Networking & Commun Chengdu 611731 Peoples R China

Spectrum and service sharing among multiple mobile network operators (MNOs) is regarded as a promising technology to construct future mobile networks, because it provides a higher utilization efficiency of the network resources, e.g., spectrum and network infrastructure. However, each individual MNO may not be able to benefit from this technology without joint optimizations. In this paper, we investigate a symbiotic spectrum and service sharing paradigm, which can improve the overall performance of the multi-MNO network while guaranteeing each MNO benefits as well. Particularly, we formulate two user association problems with novel mutual benefit constraints, namely, sum rate maximization and load balancing. Besides, we introduce a sharing coefficient for inter-MNO service sharing, which can be regarded as an inter-MNO service level agreement (SLA) accounting for the regulation of user association behaviors. Then, we propose a successive convex approximation (SCA) based algorithm and a fractional programming (FP) based algorithm to solve the sum rate maximization problems for different inter-MNO sharing strategies. In addition, we also develop a Lagrangian dual decomposition based algorithm for the load balancing problems. Finally, extensive numerical simulations are provided to demonstrate the effectiveness of the proposed algorithms. The results show that mutual benefit constraints can help multiple MNOs realize a symbiotic spectrum and service sharing paradigm. By comparing different inter-MNO sharing strategies, it can also be observed that the highest performance gain can be achieved when the multiple MNOs share their spectrum and service simultaneously.

关键词： Symbiosis Service level agreements Resource management Optimization Load management approximation algorithms Time division multiple access Symbiotic spectrum and service sharing multi-MNO network user association optimization

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An iterative approach to grid topology and redispatch optimization in congestion management

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ELECTRIC POWER SYSTEMS RESEARCH 2024年 234卷

作者： Ewerszumrode, Andrea Erle, Niklas Krahl, Simon Moser, Albert FGH eV R&D Elect Grids Aachen Germany FGH GmbH Software Solut Aachen Germany Rhein Westfal TH Aachen IAEW Aachen Germany

The selection of Remedial Actions (RA) to ensure system security is a highly complex task performed by Transmission System Operators (TSOs). Phase shifting transformer (PST) tap changes and active power changes of generation units (redispatch) are some RA available in Security Constrained Optimal Power Flow (SCOPF) simulations within the operational planning processes. Topological RA are not part of these SCOPF yet, as the optimization thereof adds high complexity to the existing optimization problem. To overcome this complexity, this paper introduces an iterative approach that decouples topology optimization from redispatch & PST optimization. Linearized models of RA are used to meet computation time requirements. Exemplary investigations of the presented method were performed based on modified IEEE 39-Bus, 118-Bus and PEGASE-1354-Bus grid models. Within the scope of these investigations, the method shows good results and a great potential for the reduction of congestions and required redispatch through topological RA. In order to eliminate inaccuracies of the approach and to further improve its suitability for use in grid operation, a need for future investigations and possible further developments were identified.

关键词： approximation algorithms Linearized models Redispatch optimization Topology optimization

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A PTAS for the horizontal rectangle stabbing problem

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MATHEMATICAL PROGRAMMING 2024年第1-2期206卷 607-630页

作者： Khan, Arindam Subramanian, Aditya Wiese, Andreas Indian Inst Sci Bengaluru India Techn Univ Munich Munich Germany

We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal-vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a (2+epsilon)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+\varepsilon )$$\end{document}-approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle's width is at most its height, and (iii) when all rectangles are delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-large, that is, have at least one edge whose length is at least delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{a

关键词： approximation algorithms Geometric optimization Rectangles Stabbing

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A New Generalized Quasi-Newton Algorithm Based on Structured Diagonal Hessian approximation for Solving Nonlinear Least-Squares Problems With Application to 3DOF Planar Robot Arm Manipulator

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IEEE ACCESS 2022年 10卷 10816-10826页

作者： Yahaya, Mahmoud Muhammad Kumam, Poom Awwal, Aliyu Muhammed Chaipunya, Parin Aji, Sani Salisu, Sani King Mongkuts Univ Technol Thonburi KMUTT KMUTTFixed Point Res Lab Dept Math Fac Sci Room SCL 802 Fixed Point LabSci Lab Bldg Bangkok 10140 Thailand King Mongkuts Univ Technol Thonburi KMUTT Dept Math Fac Sci Bangkok 10140 Thailand China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan King Mongkuts Univ Technol Thonburi KMUTT KMUTTFixed Point Res Lab Ctr Excellence Theoret & Computat Sci TaCS CoE Dept MathFac Sci Room SCL 802 Fixed Point LabSci Lab Bldg Bangkok 10140 Thailand Gombe State Univ Dept Math Fac Sci Gombe 760214 Nigeria Sule Lamido Univ Dept Math & Comp Sci Kafin Hausa 741103 Jigawa Nigeria

Many problems in science and engineering can be formulated as nonlinear least-squares (NLS) problems. Thus, the need for efficient algorithms to solve these problems can not be overemphasized. In that sense, we introduce a generalized structured-based diagonal Hessian algorithm for solving NLS problems. The formulation associated with this algorithm is essentially a generalization of a similar result in Yahaya et al. (Journal of Computational and Applied Mathematics, pp. 113582, 2021). However, in this work, the structured diagonal Hessian update is derived under a weighted Frobenius norm;this allows other choices of the weighted matrix analogous to the Davidon-Fletcher-Powell (DFP) method. Moreover, to theoretically fill the gap in Yahaya et al. (Journal of Computational and Applied Mathematics, pp. 113582, 2021), we have shown that the proposed algorithm is R-linearly convergent under some standard conditions devoid of any safeguarding strategy. Furthermore, we experimentally tested the proposed scheme on some standard benchmark problems in the literature. Finally, we applied this algorithm to solve robotic motion control problem consisting of 3DOF (degrees of freedom).

关键词： approximation algorithms Mathematics Taylor series Standards Robot motion Search problems Roads Nonlinear least squares quasi-Newton diagonal updating least change secant robotic motion control

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Joint Differential Evolution and Successive Convex approximation in UAV-Enabled Mobile Edge Computing

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IEEE ACCESS 2022年 10卷 57413-57426页

作者： Yu, Zhe Fan, Guoliang Oklahoma State Univ Sch Elect & Comp Engn Stillwater OK 74078 USA

UAV-enabled mobile edge computing (MEC) is an emerging technology to support resource-intensive yet delay-sensitive applications with edge clouds (ECs) deployed in the proximity to mobile users and UAVs served as computing base stations in the air. The formulated optimization problems therein are highly nonconvex and thus difficult to solve. To tackle the nonconvexity, the successive convex approximation (SCA) technique has been widely used to solve for the nonconvex optimization problems by transforming the nonconvex objective functions and constraints into suitable convex surrogates. However, the optimal solutions are based on the approximated optimization problem not the original one and they are highly dependent on the feasible solution initialization. Unlike SCA, Differential Evolution (DE) is a global optimization method that iteratively updates the best candidate solutions with respect to the predefined objective functions. DE works well especially in unconstrained optimization problems since it can freely search very large regions of possible solutions without considering the convexity of the original problem. However, when it comes to the constrained optimization problem, DE becomes inefficient to find the feasible and optimal solutions within given time limits. In view of the shortcomings incurred in both DE and SCA, we propose an innovative algorithm by jointly applying DE and SCA (DE-SCA) to solve for the nonconvex optimization problems. However, directly using full DE solutions to initialize the SCA-based algorithm will result in worse objective function values as the DE solutions are often infeasible. Therefore, we further design to screen the feasible parts from the DE solutions and utilize them to initialize the SCA-based algorithm. In experimental simulations, we consider a system of UAV-enabled MEC where IoT devices, the UAV and ECs interact with each other. The simulation results demonstrate that our proposed Screened DE-SCA algorithm largely outp

关键词： Optimization approximation algorithms Search problems Multi-access edge computing Linear programming Internet of Things Clustering algorithms Differential evolution mobile edge computing successive convex approximation unmanned aerial vehicles

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An O(n log n)-Time approximation Scheme for Geometric Many-To-Many Matching 40

An O(n log n)-Time Approximation Scheme for Geometric Many-T...

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40th International Symposium on Computational Geometry, SoCG 2024

作者： Bandyapadhyay, Sayan Xue, Jie Department of Computer Science Portland State University OR United States Department of Computer Science New York University Shanghai China

ISBN: (纸本)9783959773164

Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is a set S of n colored points in Rd, which implicitly defines a graph G = (S, E(S)) where E(S) = {(p, q): p, q ∈ S have different colors}, and the goal is to compute a minimum-cost subset E∗ ⊆ E(S) of edges that cover all points in S. Here the cost of E∗ is the sum of the costs of all edges in E∗, where the cost of a single edge e is the Euclidean distance (or more generally, the Lp-distance) between the two endpoints of e. Our main result is a (1 + Ε)-approximation algorithm with an optimal running time OΕ(n log n) for geometric many-to-many matching in any fixed dimension, which works under any Lp-norm. This is the first near-linear approximation scheme for the problem in any d ≥ 2. Prior to this work, only the bipartite case of geometric many-to-many matching was considered in R1 and R2, and the best known approximation scheme in R2 takes OΕ(n1.5 · poly(log n)) time. © Sayan Bandyapadhyay and Jie Xue.

关键词： approximation algorithms

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