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A convergent algorithm for bi-orthogonal nonnegative matrix tri-factorization

ADVANCES IN DATA ANALYSIS AND CLASSIFICATION 2021年第4期15卷 1069-1102页

作者： Mirzal, Andri King Fahd Univ Petr & Minerals Informat & Comp Sci Dept Dhahran Saudi Arabia

A convergent algorithm for nonnegative matrix factorization (NMF) with orthogonality constraints imposed on both basis and coefficient matrices is proposed in this paper. This factorization concept was first introduced by Ding et al. (Proceedings of 12th ACM SIGKDD international conference on knowledge discovery and data mining, pp 126-135, 2006) with intent to further improve clustering capability of NMF. However, as the original algorithm was developed based on multiplicative update rules, the convergence of the algorithm cannot be guaranteed. In this paper, we utilize the technique presented in our previous work Mirzal (J Comput Appl Math 260:149-166, 2014a;Proceedings of the first international conference on advanced data and information engineering (DaEng-2013). Springer, pp 177-184, 2014b;IEEE/ACM Trans Comput Biol Bioinform 11(6):1208-1217, 2014c) to develop a convergent algorithm for this problem and prove that it converges to a stationary point inside the solution space. As it is very hard to numerically show the convergence of an NMF algorithm due to the slow convergence and numerical precision issues, experiments are instead performed to evaluate whether the algorithm has the nonincreasing property (a necessary condition for the convergence) where it is shown that the algorithm has this property. Further, clustering capability of the algorithm is also inspected by using Reuters-21578 data corpus.

关键词： Clustering methods convergent algorithm Nonnegative matrix factorization Orthogonality constraint

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A convergent algorithm for orthogonal nonnegative matrix factorization

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 260卷 149-166页

作者： Mirzal, Andri Univ Teknol Malaysia Fac Comp Johor Baharu 81310 Malaysia

This paper proposes a convergent algorithm for nonnegative matrix factorization (NMF) with orthogonality constraint on the factors. We design the algorithm based on the additive update rule algorithm for the standard NMF proposed by Lee and Seung, and derive the convergent version by generalizing the convergence proof of the algorithm developed by Lin. Further we use the proposed algorithms to improve clustering capability of the standard NMF using the Reuter document corpus, a standard dataset in clustering research. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Nonnegative matrix factorization Orthogonality constraint convergent algorithm Clustering methods

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Reliable and efficient trajectory planning for fixed-wing UAVs via nonlinearity transfer and convexification

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AEROSPACE SCIENCE AND TECHNOLOGY 2025年 158卷

作者： Li, Yaxuan Wang, Yan Kuang, Hailiang Liu, Xinfu Beijing Inst Technol Sch Aerosp Engn Beijing 100081 Peoples R China

This paper investigates the trajectory planning problem for fixed-wing UAVs with various constraints. Solving the problem by nonlinear programming or successive convex programming often involves a high computation burden or risk of non-convergence. To enhance reliability and efficiency, we decompose the problem into two subproblems for optimizing the path and the speed, respectively. Multiple nonlinearity transfer and moderate conservative approximation are proposed to transform the former problem into a second-order cone programming problem with concave inequality constraints, which can be solved by existing methods with guaranteed convergence. The latter problem is equivalently rewritten into the same type of problem by applying variable redefinition and equivalent transformation of the objective function. An iterative algorithm of solving the two subproblems in sequence is designed to get a near-optimal solution of the original problem, and another algorithm is provided to improve the solution optimality. Rigorous analysis shows that the proposed algorithms are convergent. And numerical examples clearly demonstrate that they are much more reliable and efficient than existing algorithms.

关键词： Fixed-wing UAV Trajectory planning Convex optimization convergent algorithm Optimal control

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Minimum-time interception of a moving target by a material point in a viscous medium

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AUTOMATICA 2024年 167卷

作者： Buzikov, Maksim E. Mayer, Alina M. Russian Acad Sci VA Trapeznikov Inst Control Sci Moscow Russia

In this paper we investigated a model that describes the motion of a material point in a viscous medium under a force that is arbitrary in direction, but limited in magnitude. This model was named "the isotropic rocket" in the early work of Rufus Isaacs. We obtained a parametric description of a reachable set for the isotropic rocket and solved a group of reachability problems for a final configuration that varies in a known time-dependent manner (moving target). To describe the reachable set, we obtained an explicit parametric form of its boundary and all its projections onto all subspaces of the state space. convergent algorithms have been proposed for computing minimum-time interception in position and velocity spaces. Finally, we numerically investigated particular cases of minimum-time interception, validating the development of this study. (c) 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Isotropic rocket Moving target convergent algorithm Analytic design Time -varying systems N -dimensional systems

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Numerical solutions to singular φ-Laplacianwith Dirichlet boundary conditions

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NUMERICAL algorithmS 2014年第2期67卷 305-318页

作者： Jebelean, Petru Popa, Constantin West Univ Timisoara Dept Math Timisoara 300223 Romania Acad Romana Inst Math Simion Stoilow Bucharest Romania Ovidius Univ Constanta Fac Math & Informat Constanta 900527 Romania

In this note we are concerned with numerical solutions to Dirichlet problem [phi(u')]' = f(x) in [alpha, beta];u(alpha) = A, u(beta) = B, where phi : (-eta, eta) -> R (eta = d > 0 y is an element of(-eta... 详细信息

In this note we are concerned with numerical solutions to Dirichlet problem [phi(u')]' = f(x) in [alpha, beta];u(alpha) = A, u(beta) = B, where phi : (-eta, eta) -> R (eta < +infinity) is an increasing diffeomorphism with phi'(y) >= d > 0 y is an element of(-eta, eta) for all . The obtained algorithm combines the shooting method with Euler's method and it is convergent whenever the problem is solvable. We provide numerical experiments confirming the theoretical aspects.

关键词： Singular phi-Laplacian Shooting method Euler's method convergent algorithm Runge-Kutta method

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Orthogonal Nonnegative Matrix Factorization for Blind Image Separation

Orthogonal Nonnegative Matrix Factorization for Blind Image ...

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3rd International Visual Informatics Conference (IVIC)

作者： Mirzal, Andri Univ Teknol Malaysia Fac Comp N28 439 03 Utm Johor Bahru 81310 Malaysia

ISBN: (纸本)9783319029573;9783319029580

This paper describes an application of orthogonal nonnegative matrix factorization (NMF) algorithm in blind image separation (BIS) problem. The algorithm itself has been presented in our previous work as an attempt to provide a simple and convergent algorithm for orthogonal NMF, a type of NMF proposed to improve clustering capability of the standard NMF. When we changed the application domain of the algorithm to the BIS problem, surprisingly good results were obtained;the reconstructed images were more similar to the original ones and pleasant to view compared to the results produced by other NMF algorithms. Good results were also obtained when another dataset that consists of unrelated images was used. This practical use along with its convergence guarantee and implementation simplicity demonstrate the benefits of our algorithm.

关键词： nonnegative matrix factorization convergent algorithm blind image separation orthogonality constraint

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nVariational inequalities for solving possibilistic risk-averse electricity market equilibrium

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IET GENERATION TRANSMISSION & DISTRIBUTION 2008年第5期2卷 632-645页

作者： Campos, F. A. Villar, J. Barquin, J. Reneses, J. Univ Pontificia Comillas Madrid Inst Invest Tecnol Madrid 28015 Spain

It is widely known and accepted that Nash equilibrium suitably models agents' behavior in electricity markets, since it is coherent with the common sense of their simultaneous profits maximisation. In the literature, these approaches are usually addressed using deterministic representations, despite the fact that electricity markets are highly conditioned by the uncertainty in demand or in agents' bidding strategies. Only some equilibrium-modelling approaches under uncertainty can be found in the literature, most of them using probability distributions. However, probability approaches may lead to very complex formulations and generally require restrictive assumptions (such as normality or independence) that can hardly be verified in real complex problems. A conjectured-price-response equilibrium model that uses LR-possibility distributions to represent the uncertainty of the residual demand curves faced by the participant agents is proposed. Modelling the risk-aversion attitudes of the agents, the resulting possibilistic equilibrium is transformed into a simplified deterministic one, which is solved with a new globally convergent algorithm for variational inequalities problems. Some interesting results for a real-size electricity system show the robustness of this new approach when compared with other risk-neutral approaches.

关键词： power markets variational inequality probability risk-averse electricity market profit maximisation risk analysis convergent algorithm convergence conjectured-price-response equilibrium model pricing Nash equilibrium modelling LR-possibility distribution Power system management, operation and economics probability distribution Other topics in statistics bidding strategy

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A convergent algorithm for the output covariance constraint control problem

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1997年第1期35卷 341-361页

作者： Zhu, G Rotea, MA Skelton, R PURDUE UNIV SPACE SYST CONTROL LAB W LAFAYETTE IN 47907 USA PURDUE UNIV SCH AERONAUT & ASTRONAUT W LAFAYETTE IN 47907 USA

This paper considers the optimal control problem of minimizing control effort subject to multiple performance constraints on output covariance matrices Y-i of the form Y-i less than or equal to (Y) over bar(i), where (Y) over bar(i) is given. The contributions of this paper are a set of conditions that characterize global optimality, and an iterative algorithm for finding a solution to the optimality conditions. This iterative algorithm is completely described up to a user-specified parameter. We show that, under suitable assumptions on problem data, the iterative algorithm converges to a solution of the optimality conditions, provided that this parameter is properly chosen. Both discrete- and continuous-time problems are considered.

关键词： covariance control convergent algorithm

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The Hermite Interpolation For Nonlinear Optimization Problem

The Hermite Interpolation For Nonlinear Optimization Problem

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第三届运筹学及其应用国际研讨会

作者： Zhu Jianqing Kong Saowen Zhengzhou Institute of Surveying and Mapping

In this paper,an algorithm that uses actively constrained set and special Hermite interpolantion for the box constrained nonlinear optimization *** every step of the iteration,old iteration points serve as interpolation *** milder conditions, we prove the global convergence of the *** is also proved that the correct active set can be identified in a finite number of iteration if strict complementary slackness condition holds,and so the proposed algorithm reduces finally to an unconstrained minimization method in a limited number of iterations,allowing a fast asymptotic rate of convergence.

关键词： nonlinear oplimization problem Hermite interpolation convergent algorithm

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FAST ITERATIVE algorithm FOR RECONSTRUCTION FROM DIVERGENT-RAY PROJECTIONS

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IEE PROCEEDINGS-E COMPUTERS AND DIGITAL TECHNIQUES 1984年第3期131卷 89-96页

作者： GOUTIS, CE DROSSOS, SN Department of Electrical & Electronic Engineering Merz Laboratories University of Newcastle upon Tyne Newcastle upon Tyne UK

A convergent algorithm is introduced which solves the exact system relating the Lagrange multipliers and the projections using the successive relaxation method without any approximation. The system gives the minimum-energy solution which is different from that of the convolution algorithm, but it is the same as the ART reconstruction except that it discretises the Lagrange multipliers instead of the image. Its reconstructions are substantially better than those of the convolution algorithm. Although the new algorithm uses the exact matrix of this system, its speed is very high because it utilises the special matrix structure; namely the areas of equal elements. The statistics of the error between the ‘true’ projections and the projections corresponding to the reconstruction can be made the same as the noise statistics in the projection data, but the algorithm does not aim to produce the minimum norm solution to the problem of fitting noisy data. The number of projections N should be related to the number of measurements per projection P by P = vN/2, where v is an integer. Computed results verifying these conclusions are included. The algorithm can also use the object boundaries to improve its reconstruction.

关键词： successive relaxation method fast iterative algorithm iterative methods convergent algorithm Digital signal processing reconstruction divergent-ray projections ART reconstruction Optical information, image and video signal processing convolution algorithm computerised picture processing Communications computing Lagrange multipliers

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