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fixed point algorithm based on adapted metric method for convex minimization problem with application to image deblurring

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2016年第6期42卷 1287-1310页

作者： Chen, Dai-Qiang Zhou, Yan Song, Li-Juan Third Mil Med Univ Sch Biomed Engn Dept Math Chongqing 400038 Peoples R China Chongqing Univ Sch Automat Chongqing 401331 Peoples R China

Recently, optimization algorithms for solving a minimization problem whose objective function is a sum of two convex functions have been widely investigated in the field of image processing. In particular, the scenario when a non-differentiable convex function such as the total variation (TV) norm is included in the objective function has received considerable interests since many variational models encountered in image processing have this nature. In this paper, we propose a fast fixed point algorithm based on the adapted metric method, and apply it in the field of TV-based image deblurring. The novel method is derived from the idea of establishing a general fixed point algorithm framework based on an adequate quadratic approximation of one convex function in the objective function, in a way reminiscent of Quasi-Newton methods. Utilizing the non-expansion property of the proximity operator we further investigate the global convergence of the proposed algorithm. Numerical experiments on image deblurring problem demonstrate that the proposed algorithm is very competitive with the current state-of-the-art algorithms in terms of computational efficiency.

关键词： Adapted metric Primal-dual fixed point algorithm Total variation Image deblurring

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A VARIABLE DIMENSION fixed-point algorithm AND THE ORIENTATION OF SIMPLICES

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MATHEMATICAL PROGRAMMING 1984年第3期30卷 301-312页

作者： YAMAMOTO, Y Institute of Socio-Economic Planning University of Tsukuba 305 Sakura Ibaraki Japan

A variable dimension algorithm with integer labelling is proposed for solving systems ofn equations inn variables. The algorithm is an integer labelling version of the 2-ray algorithm proposed by the author. The orientation of lower dimensional simplices is studied and is shown to be preserved along a sequence of adjacent simplices.

关键词： fixed point algorithm System of Equations Orientation of Simplices

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THE OCTAHEDRAL algorithm, A NEW SIMPLICIAL fixed-point algorithm

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MATHEMATICAL PROGRAMMING 1981年第1期21卷 47-69页

作者： WRIGHT, AH Department of Mathematics Western Michigan University 49008 Kalamazoo Michigan USA

A new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented. It uses the restrart technique of Merrill to improve the accuracy of the solution. The algorithm is shown to converge quadratically under certain conditions. The algorithm should be efficient and relatively easy to implement.

关键词： fixed point algorithm Computation of fixed points Nonlinear Equations Variable Dimension algorithm

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A fixed point algorithm for Blind Identification of Underdetermined Mixtures

A Fixed Point Algorithm for Blind Identification of Underdet...

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22nd Chinese Control and Decision Conference

作者： Zhang, Mingjian Guangdong Univ Technol Sch Automat Guangzhou 510006 Guangdong Peoples R China

ISBN: (纸本)9781424451821

In this paper, we consider the problem of blind identification in the underdetermined instantaneous mixtures case, where there are more sources than sensors. Recently, a simultaneous matrix diagonalization (SMD)-based method has been proposed without any need of sparsity assumption on sources. Instead of using SMD technique, we propose a fixed point algorithm, which estimates the columns of the mixing matrix sequentially one by one. Computer simulations are presented in order to demonstrate that the proposed algorithm has the ability to identify the mixing matrix, and its performance is comparable to that of SMD-based algorithm.

关键词： Blind identification fixed point algorithm underdetermined mixtures parallel factor (PARAFAC) analysis simultaneous matrix diagonalization (SMD)

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A fixed point algorithm for Blind Identification of Underdetermined Mixtures

A Fixed Point Algorithm for Blind Identification of Underdet...

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The 22nd China Control and Decision Conference(2010年中国控制与决策会议)

作者： Mingjian Zhang School of Automation Guangdong University of Technology Guangzhou 510006

关键词： Blind identification fixed point algorithm underdetermined mixtures parallel factor (PARAFAC) analysis simultaneous matrix diagonalization (SMD)

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A fixed point algorithm for Blind Identification of Underdetermined Mixtures

A Fixed Point Algorithm for Blind Identification of Underdet...

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2010 Chinese Control and Decision Conference

作者： Mingjian Zhang School of Automation,Guangdong University of Technology,Guangzhou,510006

In this paper,we consider the problem of blind identification in the underdetermined instantaneous mixtures case,where there are more sources than ***,a simultaneous matrix diagonalization(SMD)-based method has been proposed without any need of sparsity assumption on *** of using SMD technique,we propose a fixed point algorithm,which estimates the columns of the mixing matrix sequentially one by *** simulations are presented in order to demonstrate that the proposed algorithm has the ability to identify the mixing matrix,and its performance is comparable to that of SMD-based algorithm.

关键词： Blind identification fixed point algorithm underdetermined mixtures parallel factor(PARAFAC) analysis simultaneous matrix diagonalization(SMD)

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A fixed point approach for computing actuarially fair Pareto optimal risk-sharing rules

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EUROPEAN ACTUARIAL JOURNAL 2025年第1期15卷 297-334页

作者： Niakh, Fallou Ctr Rech Econ & Stat CREST Inst Polytech Paris Ecole Polytech CNRSGENESENSAE Paris F-91120 Palaiseau France

Risk-sharing is one way to pool risks without the need for a third party. To ensure the attractiveness of such a system, the rule should be accepted and understood by all participants. A desirable risk-sharing rule should fulfill actuarial fairness and Pareto optimality while being easy to compute. This paper establishes a one-to-one correspondence between an actuarially fair Pareto optimal (AFPO) risk-sharing rule and a fixed point of a specific function. A fast numerical method for computing these risk-sharing rules is also derived. As a result, we can compute AFPO risk-sharing rules for a large number of heterogeneous participants in this framework.

关键词： Pareto optimal risk-sharing rules Actuarial fairness fixed point algorithm

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SMT-based and fixed-point approaches for state estimation in max-plus linear systems

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DISCRETE EVENT DYNAMIC SYSTEMS-THEORY AND APPLICATIONS 2025年 1-17页

作者： Espindola-Winck, Guilherme Hardouin, Laurent Lhommeau, Mehdi Univ Lille CNRS Cent Lille UMR 9189CRIStAL F-59000 Lille France Univ Angers LARIS SFR MATHST F-49000 Angers France

This work builds on the seminal paper (Winck et al. IFAC-PapersOnLine 58(1):36-41 2024) and evaluates an existing method against a new approach for state estimation in Max-Plus Linear systems with bounded uncertainties. Traditional stochastic filtering is inapplicable to this system class, even though the posterior probability density function (PDF) can be computed. Previous research has shown a limited scalability of the disjunctive approach using difference-bound matrices. To address this, we investigate an alternative method recently explored in Mufid et al. (IFAC-PapersOnLine 53(4):459-465 2020, IEEE Trans Autom Control 67(6):2700-2714 2022), employing Satisfiability Modulo Theory (SMT) techniques, despite their NP-hard nature. The main novelty of this work is the proposal of a concise method based on fixed-point iteration in max-plus algebra, which is known to be a pseudo-polynomial time algorithm. To compare both approaches, a representative autonomous system is used in the paper to illustrate the basic computations. The efficiency of both approaches is compared through numerical experiments.

关键词： Max-plus linear systems State-estimation Satisfiability modulo theories fixed point algorithm

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Dynamic traffic assignment for electric vehicles

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TRANSPORTATION RESEARCH PART B-METHODOLOGICAL 2025年 195卷

作者： Graf, Lukas Harks, Tobias Palkar, Prashant Augsburg Univ Inst Math D-86135 Augsburg Germany Univ Passau Fac Comp Sci & Math D-94032 Passau Germany Indian Inst Technol Delhi Dept Mech Engn New Delhi 110016 India

We initiate the study of dynamic traffic assignment for electrical vehicles addressing the specific challenges such as range limitations and the possibility of battery recharge at predefined charging locations. As our main result, we establish the existence of energy-feasible dynamic equilibria within networks using the deterministic queuing model of Vickrey for the flow dynamics on edges. There are three key modeling-ingredients for obtaining this existence result: 1. We introduce a walk-based definition of dynamic traffic flows which allows for cyclic routing behavior as a result of recharging events en route. 2. We use abstract convex feasibility sets in an appropriate function space to model the energy-feasibility of used walks. 3. We introduce the concept of capacitated dynamic equilibrium walk-flows which generalize the former unrestricted dynamic equilibrium path-flows. Viewed in this framework, we show the existence of an energy-feasible dynamic equilibrium by applying an infinite dimensional variational inequality, which in turn requires a careful analysis of continuity properties of the network loading as a result of injecting flow into walks. We complement our theoretical results by a computational study in which we design a fixed-point algorithm computing energy-feasible dynamic equilibria. We apply the algorithm to standard real-world instances from the traffic assignment community. The study demonstrates that battery constraints have a significant impact on the resulting travel times and energy consumption profiles compared to conventional fuel-based vehicles. We further show that our algorithm computes (approximate) equilibria for small and medium sized instances in acceptable running times but struggles for larger instances.

关键词： Electromobility Dynamic traffic assignment Dynamic flows fixed point algorithm

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Global Convergence and Acceleration of Projection Methods for Feasibility Problems Involving Union Convex Sets

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2025年第2期204卷 1-37页

作者： Alcantara, Jan Harold Lee, Ching-pei RIKEN Ctr Adv Intelligence Project Chuo Ku Tokyo Japan Inst Stat Math Dept Adv Data Sci Tachikawa Tokyo Japan

We prove global convergence of classical projection algorithms for feasibility problems involving union convex sets, which refer to sets expressible as the union of a finite number of closed convex sets. We present a unified strategy for analyzing global convergence by means of studying fixed-point iterations of a set-valued operator that is the union of a finite number of compact-valued upper semicontinuous maps. Such a generalized framework permits the analysis of a class of proximal algorithms for minimizing the sum of a piecewise smooth function and the difference between the pointwise minimum of finitely many weakly convex functions and a piecewise smooth convex function. When realized on two-set feasibility problems, this algorithm class recovers alternating projections and averaged projections as special cases, and thus we obtain global convergence criterion for these projection algorithms. Using these general results, we derive sufficient conditions to guarantee global convergence for several projection algorithms for solving the sparse affine feasibility problem and a feasibility reformulation of the linear complementarity problem. Notably, we obtain global convergence of both the alternating and the averaged projection methods to the solution set for linear complementarity problems involving P-matrices. By leveraging the structures of the classes of problems we consider, we also propose acceleration algorithms with guaranteed global convergence. Numerical results further exemplify that the proposed acceleration schemes significantly improve upon their non-accelerated counterparts in efficiency.

关键词： fixed point algorithm Proximal methods Alternating projections Averaged projections Linear complementarity problem Union convex set Nonconvex feasibility problems Nonconvex optimization Global convergence

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