103 results
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2. WEAK AND LINEAR CONVERGENCE OF A GENERALIZED PROXIMAL POINT ALGORITHM WITH ALTERNATING INERTIAL STEPS FOR A MONOTONE INCLUSION PROBLEM.
- Author
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YEKINI SHEHU and EZEORA, JEREMIAH N.
- Subjects
STOCHASTIC convergence ,ALGORITHMS ,PROBLEM solving ,MATHEMATICAL analysis ,LINEAR statistical models - Abstract
The proximal point algorithm (PPA) is a powerful tool for solving monotone inclusion problems. Recently, Tao and Yuan [On the optimal linear convergence rate of a generalized proximal point algorithm, J. Sci. Comput. 74 (2018), 826-850] proposed a generalized PPA (GPPA) for finding a zero point of a maximal monotone operator, and obtained the linear convergence rate of the generalized PPA. In this paper, we consider accelerating the GPPA with the aid of the inertial extrapolation. We propose a generalized proximal point algorithm with alternating inertial steps solving monotone inclusion problem, and obtain weak convergence results under some mild conditions. When the inverse of the involved monotone operator is Lipschitz continuous at the origin, we prove that the iterative sequence generated by our generalized proximal point algorithm is linearly convergent. The Fej'er monotonicity of even subsequences of the iterative sequence is also recovered. Finally, we give some priori and posteriori error estimates of our generated sequences. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
3. A NEW ITERATIVE METHOD WITH ALTERNATED INERTIA FOR THE SPLIT FEASIBILITY PROBLEM.
- Author
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QIAO-LI DONG, LULU LIU, LUNLONG ZHONG, and DONGLI ZHANG
- Subjects
ITERATIVE methods (Mathematics) ,STOCHASTIC convergence ,ALGORITHMS ,PROBLEM solving ,INFORMATION retrieval - Abstract
In this paper, an iterative method with alternated inertial extrapolation step is proposed to solve the split feasibility problem. The stepsize in the proposed algorithm uses the self adaptive technique, which does not depend on the prior information of the operator norm. The weak convergence is proved under suitable conditions. Finally, a numerical example is given to illustrate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
4. On the Theoretical Analysis of the Plant Propagation Algorithms.
- Author
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Sulaiman, Muhammad, Salhi, Abdellah, Khan, Asfandyar, Muhammad, Shakoor, and Khan, Wali
- Subjects
MATHEMATICAL optimization ,ALGORITHMS ,HEURISTIC algorithms ,PROBLEM solving ,STOCHASTIC convergence - Abstract
Plant Propagation Algorithms (PPA) are powerful and flexible solvers for optimisation problems. They are nature-inspired heuristics which can be applied to any optimisation/search problem. There is a growing body of research, mainly experimental, on PPA in the literature. Little, however, has been done on the theoretical front. Given the prominence this algorithm is gaining in terms of performance on benchmark problems as well as practical ones, some theoretical insight into its convergence is needed. The current paper is aimed at fulfilling this by providing a sketch for a global convergence analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
5. Restricted Normal Cones and Sparsity Optimization with Affine Constraints.
- Author
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Bauschke, Heinz, Luke, D., Phan, Hung, and Wang, Xianfu
- Subjects
MATHEMATICAL optimization ,PROBLEM solving ,PARALLEL computers ,STOCHASTIC convergence ,ALGORITHMS ,ESTIMATION theory - Abstract
The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted normal cone which generalizes the classical Mordukhovich normal cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted normal cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
6. General split equality problems in Hilbert spaces.
- Author
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Rudong Chen, Jie Wang, and Huiwen Zhang
- Subjects
HILBERT space ,PROBLEM solving ,CONVEX functions ,ALGORITHMS ,STOCHASTIC convergence ,DIMENSIONAL analysis - Abstract
A new convex feasibility problem, the split equality problem (SEP), has been proposed by Moudafi and Byrne. The SEP was solved through the ACQA and ARCQA algorithms. In this paper the SEPs are extended to infinite-dimensional SEPs in Hilbert spaces and we established the strong convergence of a proposed algorithm to a solution of general split equality problems (GSEPs). [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
7. An improved first-order primal-dual algorithm with a new correction step.
- Author
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Cai, Xingju, Han, Deren, and Xu, Lingling
- Subjects
CORRECTION factors ,STOCHASTIC convergence ,ALGORITHMS ,NUMERICAL analysis ,IMAGE reconstruction ,PROBLEM solving - Abstract
In this paper, we propose a new correction strategy for some first-order primal-dual algorithms arising from solving, e.g., total variation image restoration. With this strategy, we can prove the convergence of the algorithm under more flexible conditions than those proposed most recently. Some preliminary numerical results of image deblurring support that the new correction strategy can improve the numerical efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
8. Geometric branch-and-bound methods for constrained global optimization problems.
- Author
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Scholz, Daniel
- Subjects
GLOBAL optimization ,ALGORITHMS ,PROBLEM solving ,MATHEMATICAL proofs ,STOCHASTIC convergence ,NUMERICAL analysis - Abstract
Geometric branch-and-bound methods are popular solution algorithms in deterministic global optimization to solve problems in small dimensions. The aim of this paper is to formulate a geometric branch-and-bound method for constrained global optimization problems which allows the use of arbitrary bounding operations. In particular, our main goal is to prove the convergence of the suggested method using the concept of the rate of convergence in geometric branch-and-bound methods as introduced in some recent publications. Furthermore, some efficient further discarding tests using necessary conditions for optimality are derived and illustrated numerically on an obnoxious facility location problem. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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9. Finite element analysis of two- and three-dimensional static problems in the asymmetric theory of elasticity as a basis for the design of experiments.
- Author
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Korepanov, V., Matveenko, V., and Shardakov, I.
- Subjects
FINITE element method ,ELASTICITY ,EXPERIMENTAL design ,PROBLEM solving ,RELIABILITY (Personality trait) ,ALGORITHMS ,STOCHASTIC convergence ,ESTIMATION theory - Abstract
In this paper, the constitutive relations of the finite element method are constructed and used for solving two- and three-dimensional problems of the asymmetric theory of elasticity. Different variants of finite elements are considered. The numerical experiments are carried out to evaluate the reliability and computational efficiency of the finite element algorithm based on the comparison between the numerical and analytical solutions, numerical estimation of the convergence and checking of the degree of accuracy, to which the natural boundary conditions are satisfied. The obtained solutions to the two- and three-dimensional problems are interpreted from the viewpoint of their applicability to a design of experiments capable of revealing the facts of couple-stress effects in material under elastic deformation and identification of material constants for the asymmetric theory of elasticity. The capabilities of the finite element algorithm to interpret experimental data and estimate the errors occurring in real experiments have been tested by solving several example problems. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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10. A Proximal Analytic Center Cutting Plane Algorithm for Solving Variational Inequality Problems.
- Author
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Jie Shen and Pang, Li-Ping
- Subjects
ALGORITHMS ,VARIATIONAL inequalities (Mathematics) ,MATHEMATICAL mappings ,APPROXIMATION theory ,STOCHASTIC convergence ,PROBLEM solving ,MATHEMATICAL analysis - Abstract
Under the condition that the values of mapping F are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
11. On Convergence of the MC Algorithm for Subspace Computation.
- Author
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Hua, Yingbo and Tianping Chen
- Subjects
ALGORITHMS ,STOCHASTIC convergence ,TOOLS ,METHODOLOGY ,ELECTRONIC data processing ,PROBLEM solving - Abstract
The "novel information criterion" (NIC) algorithm was developed by Miao and Hua in 1998 for fast adaptive computation of the principal subspace of a vector sequence. The MC algorithm is as efficient computationally as the PAST method, which was devised by Yang in 1995, and also has an attractive orthonormal property. Although all available evidence suggests that the NIC algorithm converges to the desired solution for any fixed leakage factor between zero and one, a complete proof (or disproof) has not been found, except for an arbitrarily small leakage factor. This paper presents this tong-standing open problem with a discussion of what is known so far. The results shown in this paper provide a new insight into the orthonormal property of the MC algorithm at convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
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12. A SUPERLINEARLY CONVERGENT ALGORITHM FOR THE MONOTONE NONLINEAR COMPLEMENTARITY PROBLEM WITHOUT UNIQUENESS AND NONDEGENERACY CONDITIONS.
- Author
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Dan, Hiroshige, Yamashita, Nobuo, and Fukushima, Masao
- Subjects
ALGORITHMS ,NONLINEAR theories ,STOCHASTIC convergence ,LINEAR complementarity problem ,OPERATOR theory ,MATHEMATICAL sequences ,NUMERICAL solutions to equations ,PROBLEM solving - Abstract
The purpose of this paper is to present an algorithm for solving the monotone nonlinear complementarity problem (NCP) that enjoys superlinear convergence in a genuine sense without the uniqueness and nondegeneracy conditions. Recently, Yamashita and Fukushima (2001) proposed a method based on the proximal point algorithm (PPA) for monotone NCP. The method has the favorable property that a generated sequence converges to the solution set of NCP superlinearly. However, when a generated sequence converges to a degenerate solution, the subproblems may become computationally expensive and the method does not have genuine superlinear convergence. More recently, Yamashita et al. (2001) presented a technique to identify whether a solution is degenerate or not. Using this technique, we construct a differentiable system of nonlinear equations in which the solution is a solution of the original NCP. Moreover, we propose a hybrid algorithm that is based on the PPA and uses this system. We show that the proposed algorithm has a genuine quadratic or superlinear rate of convergence even if it converges to a solution that is neither unique nor nondegenerate. [ABSTRACT FROM AUTHOR]
- Published
- 2002
- Full Text
- View/download PDF
13. On the convergence of two-dimensional fuzzy Volterra-Fredholm integral equations by using Picard method.
- Author
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Ebadian, Ali, Farahrooz, Foroozan, and Khajehnasiri, Amirahmad
- Subjects
STOCHASTIC convergence ,PICARD groups ,FUZZY mathematics ,ALGORITHMS ,VOLTERRA equations ,FREDHOLM equations ,PROBLEM solving - Abstract
In this paper we prove convergence of the method of successive approximations used to approximate the solution of nonlinear two-dimensional Volterra-Fredholm integral equations and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration. Also we present an iterative procedure to solve such equations. Finally, the method is illustrated by solving some examples. [ABSTRACT FROM AUTHOR]
- Published
- 2016
14. A nonmonotone flexible filter method for nonlinear constrained optimization.
- Author
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Su, Ke, Li, Xiaochuan, and Hou, Ruyue
- Subjects
NONLINEAR theories ,FILTERS (Mathematics) ,CONSTRAINED optimization ,PROBLEM solving ,ALGORITHMS ,STOCHASTIC convergence - Abstract
In this paper, we present a flexible nonmonotone filter method for solving nonlinear constrained optimization problems which are common models in industry. This new method has more flexibility for the acceptance of the trial step compared to the traditional filter methods, and requires less computational costs compared with the monotone-type methods. Moreover, we use a self-adaptive parameter to adjust the acceptance criteria, so that Maratos effect can be avoided a certain degree. Under reasonable assumptions, the proposed algorithm is globally convergent. Numerical tests are presented that confirm the efficiency of the approach. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
15. A CLASS OF INSIDE-OUT ALGORITHMS FOR GENERAL PROGRAMS.
- Author
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Gould, F. J.
- Subjects
ALGORITHMS ,STOCHASTIC convergence ,NONLINEAR programming ,MATHEMATICAL programming ,MATHEMATICAL optimization ,GEOMETRICAL constructions ,MANAGEMENT science ,MATHEMATICAL models ,MATHEMATICAL analysis ,OPERATIONS research ,MATHEMATICAL functions ,PROBLEM solving - Abstract
In this paper the Fiacco-McCormick SUMT technique is embedded in a class of inside-out algorithms. Convergence is demonstrated for the nonlinear programming problem under fairly general conditions and the algorithms are interpreted in a geometric structure. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
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16. Implementation of Train Scheduling System in Rail Transport using Assignment Problem Solution.
- Author
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Pashchenko, F.F., Kuznetsov, N.A., Ryabykh, N.G., Minashina, I.K., Zakharova, E.M., and Tsvetkova, O.A.
- Subjects
TRAIN schedules ,PRODUCTION scheduling ,ASSIGNMENT problems (Programming) ,PROBLEM solving ,ALGORITHMS ,STOCHASTIC convergence - Abstract
The paper is concerned with the freight scheduling problem in the rail transport control systems. Its base sub problem of train scheduling has been modeled as an assignment problem and solved using auction method. Results of comparison of auction algorithm and Hungarian algorithm application for train scheduling problem showed that auction algorithm is significantly faster in convergence. Towards the end, in this paper, was proposed the ways of further algorithm improvement. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
17. A hybrid algorithm based on particle swarm and chemical reaction optimization for multi-object problems.
- Author
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Li, Zhiyong, Nguyen, Tien Trong, Chen, ShaoMiao, and Truong, Tung Khac
- Subjects
PARTICLE swarm optimization ,ALGORITHMS ,CHEMICAL reactions ,MULTIDISCIPLINARY design optimization ,PROBLEM solving ,STOCHASTIC convergence - Abstract
Over the past decade, the particle swarm optimization (PSO) has been an effective algorithm for solving single and multi-object optimization problems. Recently, the chemical reaction optimization (CRO) algorithm is emerging as a new algorithm used to efficiently solve single-object optimization. In this paper, we present HP-CRO (hybrid of PSO and CRO) a new hybrid algorithm for multi-object optimization. This algorithm has features of CRO and PSO, HP-CRO creates new molecules (particles) not only used by CRO operations as found in CRO algorithm but also by mechanisms of PSO. The balancing of CRO and PSO operators shows that the method can be used to avoid premature convergence and explore more in the search space. This paper proposes a model with modified CRO operators and also adding new saving molecules into the external population to increase the diversity. The experimental results of the HP-CRO algorithm compared to some meta-heuristics algorithms such as FMOPSO, MOPSO, NSGAII and SPEA2 show that there is improved efficiency of the HP-CRO algorithm for solving multi-object optimization problems. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
18. Strong Convergence for the Split Common Fixed-Point Problem for Total Quasi-Asymptotically Nonexpansive Mappings in Hilbert Space.
- Author
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Mohammed, Lawan Bulama and Kılıçman, A.
- Subjects
- *
STOCHASTIC convergence , *FIXED point theory , *PROBLEM solving , *NONEXPANSIVE mappings , *HILBERT space , *ALGORITHMS - Abstract
In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
19. A Noninterior Path following Algorithm for Solving a Class of Multiobjective Programming Problems.
- Author
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Menglong Su, Zhengbang Zha, and Zhonghai Xu
- Subjects
- *
PROBLEM solving , *NUMERICAL analysis , *AUTOMOTIVE engineering , *STOCHASTIC convergence , *ALGORITHMS - Abstract
Multi objective programming problems have been widely applied to various engineering areas which include optimal design of an automotive engine, economics, and military strategies. In this paper, we propose a noninterior path following algorithm to solve a class of multi objective programming problems. Under suitable conditions, a smooth path will be proven to exist. This can give a constructive proof of the existence of solutions and lead to an implementable globally convergent algorithm. Several numerical examples are given to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
20. Algorithms for Solving the Variational Inequality Problem over the Triple Hierarchical Problem.
- Author
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Jitpeera, Thanyarat and Kumam, Poom
- Subjects
VARIATIONAL inequalities (Mathematics) ,ALGORITHMS ,PROBLEM solving ,MONOTONE operators ,STOCHASTIC convergence ,FIXED point theory - Abstract
This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
21. Variant Gradient Projection Methods for the Minimization roblems.
- Author
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Yonghong Yao, Yeong-Cheng Liou, and Ching-Feng Wen
- Subjects
STOCHASTIC convergence ,GRAPHICAL projection ,PROBLEM solving ,ALGORITHMS ,HILBERT space - Abstract
The gradient projection algorithm plays an important role in solving constrained convex minimization problems. In general, the gradient projection algorithm has only weak convergence in infinite-dimensional Hilbert spaces. Recently, H. K. Xu (2011) provided two modified gradient projection algorithms which have strong convergence. Motivated by Xu's work, in the present paper, we suggest three more simpler variant gradient projection methods so that strong convergence is guaranteed. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
22. On Variational Inclusion and Common Fixed Point Problems in q-Uniformly Smooth Banach Spaces.
- Author
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Yanlai Song, Huiying Hu, and Luchuan Ceng
- Subjects
DIFFERENTIAL inclusions ,FIXED point theory ,ITERATIVE methods (Mathematics) ,ALGORITHMS ,SET theory ,BANACH spaces ,STOCHASTIC convergence ,PROBLEM solving - Abstract
We introduce a general iterative algorithm for finding a common element of the common fixedpoint set of an infinite family of ?
i -strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent. [ABSTRACT FROM AUTHOR]- Published
- 2012
- Full Text
- View/download PDF
23. A Hybrid Extragradient-Like Method for Variational Inequalities, Equilibrium Problems, and an Infinitely Family of Strictly Pseudocontractive Mappings.
- Author
-
Yaqin Wang and Xiaoli Fang
- Subjects
VARIATIONAL inequalities (Mathematics) ,PROBLEM solving ,CONTRACTIONS (Topology) ,MATHEMATICAL mappings ,STOCHASTIC convergence ,HILBERT space ,ITERATIVE methods (Mathematics) ,ALGORITHMS - Abstract
The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan 2010 and many others. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
24. New Nonsmooth Equations-Based Algorithms for l₁-Norm Minimization and Applications.
- Author
-
Lei Wu and Zhe Sun
- Subjects
NONSMOOTH optimization ,ALGORITHMS ,PROBLEM solving ,ALGEBRAIC equations ,STOCHASTIC convergence ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
Recently, Xiao et al. proposed a nonsmooth equations-based method to solve the l₁-norm minimization problem (2011). The advantage of this method is its simplicity and lower storage. In this paper, based on new nonsmooth equations reformulation, we investigate new nonsmooth equations-based algorithms for solving l₁-norm minimization problems. Under mild conditions, we show that the proposed algorithms are globally convergent. The preliminary numerical results demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
25. Lattice rules with random [formula omitted] achieve nearly the optimal [formula omitted] error independently of the dimension.
- Author
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Kritzer, Peter, Kuo, Frances Y., Nuyens, Dirk, and Ullrich, Mario
- Subjects
- *
LATTICE field theory , *ALGORITHMS , *NUMERICAL integration , *STOCHASTIC convergence , *PROBLEM solving - Abstract
Abstract We analyze a new random algorithm for numerical integration of d -variate functions over [ 0 , 1 ] d from a weighted Sobolev space with dominating mixed smoothness α ≥ 0 and product weights 1 ≥ γ 1 ≥ γ 2 ≥ ⋯ > 0 , where the functions are continuous and periodic when α > 1 ∕ 2. The algorithm is based on rank-1 lattice rules with a random number of points n. For the case α > 1 ∕ 2 , we prove that the algorithm achieves almost the optimal order of convergence of O (n − α − 1 ∕ 2) , where the implied constant is independent of the dimension d if the weights satisfy ∑ j = 1 ∞ γ j 1 ∕ α < ∞. The same rate of convergence holds for the more general case α > 0 by adding a random shift to the lattice rule with random n. This shows, in particular, that the exponent of strong tractability in the randomized setting equals 1 ∕ (α + 1 ∕ 2) , if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-1 lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
26. A New Branch and Bound Method for Solving Sum of Linear Ratios Problem.
- Author
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Chun-Feng Wang and Xin-Yue Chu
- Subjects
- *
MATHEMATICAL bounds , *ALGORITHMS , *STOCHASTIC convergence , *LINEAR programming , *PROBLEM solving - Abstract
For globally solving sum of linear ratios problem (SLRP), this paper presents a new branch-and-bound method. In this method, a new linear relaxation technique is proposed firstly; then, the initial problem SLRP is solved by a sequence of linear programming problems. Meanwhile, to improve the convergence speed of our algorithm, two accelerating techniques are presented. The proposed algorithm is proved to be convergent, and some experiments are provided to show its feasibility and efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2017
27. A New Branch and Reduce Approach for Solving Generalized Linear Fractional Programming.
- Author
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Yong-Hong Zhang and Chun-Feng Wang
- Subjects
- *
FRACTIONAL programming , *MATHEMATICAL sequences , *ALGORITHMS , *STOCHASTIC convergence , *PROBLEM solving - Abstract
In this paper, for solving generalized linear fractional programming (GLFP), a new branch-and-reduce approach is presented. Firstly, an equivalent problem (EP) of GLFP is given; then, a new linear relaxation technique is proposed; finally, the problem EP is reduced to a sequence of linear programming problems by using the new linear relaxation technique. Meanwhile, to improve the convergence speed of our algorithm, two reducing techniques are presented. The proposed algorithm is proved to be convergent, and some experiments are provided to show its feasibility and efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2017
28. THE CONVEX SIMPLEX METHOD.
- Author
-
Zangwill, Willard I.
- Subjects
CONVEX functions ,LINEAR systems ,SIMPLEXES (Mathematics) ,TRANSPORTATION problems (Programming) ,PROBLEM solving ,MANAGEMENT education ,ALGORITHMS ,MATHEMATICAL models ,STOCHASTIC convergence ,EDUCATION - Abstract
This paper presents a method, called the convex simplex method, for minimizing a convex objective function subject to linear inequality constraints. The method is a true generalization of Dantzig's linear simplex method both in spirit and in the fact that the same tableau and variable selection techniques are used. With a linear objective function the convex simplex method reduces to the linear simplex method. Moreover, the convex simplex method actually behaves like the linear simplex method whenever it encounters a linear portion of a convex objective function. Many of the sophisticated techniques designed to enhance the efficiency of the linear simplex method are applicable to the convex simplex method. In particular, as an example, a network transportation problem with a convex objective function is solved by using the standard transportation tableau and by only slightly modifying the usual procedure for a linear objective function. [ABSTRACT FROM AUTHOR]
- Published
- 1967
- Full Text
- View/download PDF
29. Relaxed alternating CQ algorithms for the split equality problem in Hilbert spaces.
- Author
-
Yu, Hai and Wang, Fenghui
- Subjects
ALGORITHMS ,PROBLEM solving ,HILBERT space ,CONVEX functions ,STOCHASTIC convergence - Abstract
In this paper, we are concerned with the split equality problem (SEP) in Hilbert spaces. By converting it to a coupled fixed-point equation, we propose a new algorithm for solving the SEP. Whenever the convex sets involved are level sets of given convex functionals, we propose two new relaxed alternating algorithms for the SEP. The first relaxed algorithm is shown to be weakly convergent and the second strongly convergent. A new idea is introduced in order to prove strong convergence of the second relaxed algorithm, which gives an affirmative answer to Moudafi's question. Finally, preliminary numerical results show the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
30. Convergence Analysis of an Inexact Three-Operator Splitting Algorithm.
- Author
-
Zong, Chunxiang, Tang, Yuchao, and Cho, Yeol Je
- Subjects
STOCHASTIC convergence ,OPERATOR theory ,PROBLEM solving ,ALGORITHMS ,HILBERT space - Abstract
The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
31. Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term.
- Author
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Hu, Jiahui, Wang, Jungang, and Nie, Yufeng
- Subjects
ALGORITHMS ,WAVE equation ,FRACTIONAL differential equations ,STOCHASTIC convergence ,NONLINEAR analysis ,PROBLEM solving - Abstract
Fractional differential equations (FDEs) of distributed-order are important in depicting the models where the order of differentiation distributes over a certain range. Numerically solving this kind of FDEs requires not only discretizations of the temporal and spatial derivatives, but also approximation of the distributed-order integral, which brings much more difficulty. In this paper, based on the mid-point quadrature rule and composite two-point Gauss-Legendre quadrature rule, two finite difference schemes are established. Different from the previous works, which concerned only one- or two-dimensional problems with linear source terms, time-fractional wave equations of distributed-order whose source term is nonlinear in two and even three dimensions are considered. In addition, to improve the computational efficiency, the technique of alternating direction implicit (ADI) decomposition is also adopted. The unique solvability of the difference scheme is discussed, and the unconditional stability and convergence are analyzed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithms for both the two- and three-dimensional cases. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Ions motion algorithm for solving optimization problems.
- Author
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Javidy, Behzad, Hatamlou, Abdolreza, and Mirjalili, Seyedali
- Subjects
ALGORITHMS ,MATHEMATICAL optimization ,PROBLEM solving ,STOCHASTIC convergence ,MATHEMATICAL functions - Abstract
This paper proposes a novel optimization algorithm inspired by the ions motion in nature. In fact, the proposed algorithm mimics the attraction and repulsion of anions and cations to perform optimization. The proposed algorithm is designed in such a way to have the least tuning parameters, low computational complexity, fast convergence, and high local optima avoidance. The performance of this algorithm is benchmarked on 10 standard test functions and compared to four well-known algorithms in the literature. The results demonstrate that the proposed algorithm is able to show very competitive results and has merits in solving challenging optimization problems. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
33. Variable Step-Size Method Based on a Reference Separation System for Source Separation.
- Author
-
Xu, Pengcheng, Yuan, Zhigang, Jian, Wei, and Zhao, Wei
- Subjects
SIGNAL separation ,PROBLEM solving ,STOCHASTIC convergence ,MEAN square algorithms ,ALGORITHMS - Abstract
Traditional variable step-size methods are effective to solve the problem of choosing step-size in adaptive blind source separation process. But the initial setting of learning rate is vital, and the convergence speed is still low. This paper proposes a novel variable step-size method based on reference separation system for online blind source separation. The correlation between the estimated source signals and original source signals increases along with iteration. Therefore, we introduce a reference separation system to approximately estimate the correlation in terms of mean square error (MSE), which is utilized to update the step-size. The use of “minibatches” for the computation of MSE can reduce the complexity of the algorithm to some extent. Moreover, simulations demonstrate that the proposed method exhibits superior convergence and better steady-state performance over the fixed step-size method in the noise-free case, while converging faster than classical variable step-size methods in both stationary and nonstationary environments. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
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34. Global Minimization for Generalized Polynomial Fractional Program.
- Author
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Xue-Ping Hou, Pei-Ping Shen, and Chun-Feng Wang
- Subjects
- *
GLOBAL optimization , *PROBLEM solving , *ALGORITHMS , *POLYNOMIALS , *STOCHASTIC convergence , *FRACTIONAL programming - Abstract
This paper is concerned with an efficient global optimization algorithm for solving a kind of fractional program problem (P), whose objective and constraints functions are all defined as the sum of ratios generalized polynomial functions. The proposed algorithm is a combination of the branch-and-bound search and two reduction operations, based on an equivalent monotonic optimization problem of (P). The proposed reduction operations specially offer a possibility to cut away a large part of the currently investigated region in which the global optimal solution of (P) does not exist, which can be seen as an accelerating device for the solution algorithm of (P). Furthermore, numerical results show that the computational efficiency is improved by using these operations in the number of iterations and the overall execution time of the algorithm, compared with other methods. Additionally, the convergence of the algorithm is presented, and the computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem (P) provided that the number of variables is not too large. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
35. The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems.
- Author
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Najafi, H. Saberi, Edalatpanah, S. A., and Shahabi, S.
- Subjects
FUZZY systems ,LINEAR systems ,STOCHASTIC convergence ,ITERATIVE methods (Mathematics) ,ALGORITHMS ,PROBLEM solving - Abstract
We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
36. The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis.
- Author
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Chen, Zhongming, Qi, Liqun, Yang, Qingzhi, and Yang, Yuning
- Subjects
- *
EIGENVALUES , *NONNEGATIVE matrices , *TENSOR algebra , *STOCHASTIC convergence , *PROBLEM solving , *SINGULAR value decomposition , *ALGORITHMS - Abstract
Abstract: In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
37. Hybrid Estimation of Distribution Algorithm for the Quay Crane Scheduling Problem.
- Author
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Expósito-Izquierdo, Christopher, González-Velarde, José Luis, Melián-Batista, Belén, and Marcos Moreno-Vega, J.
- Subjects
ESTIMATION theory ,ALGORITHMS ,PROBLEM solving ,ROBUST control ,STOCHASTIC convergence ,COMPARATIVE studies - Abstract
Abstract: The competitiveness of a container terminal is highly conditioned by the time that container vessels spend on it. The proper scheduling of the quay cranes can reduce this time and allows a container terminal to be more attractive to shipping companies. The goal of the Quay Crane Scheduling Problem (QCSP) is to minimize the handling time of the available quay cranes when performing the tasks of loading and unloading containers onto/from a container vessel. This paper proposes a hybrid Estimation of Distribution Algorithm with local search to solve the QCSP. This approach includes a priori knowledge about the problem in the initialization step to reach promising regions of the search space as well as a novel restarting strategy with the aim of avoiding the premature convergence of the search. Furthermore, an approximate evaluation scheme is applied in order to reduce the computational burden. Moreover, its performance is statistically compared with the best optimization method from the literature. Numerical testing results demonstrate the high robustness and efficiency of the developed technique. Additionally, some relevant components of the scheme are individually analyzed to check their effectiveness. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
38. Methods for Solving Generalized Nash Equilibrium.
- Author
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Biao Qu and Jing Zhao
- Subjects
- *
GENERALIZATION , *NASH equilibrium , *MATHEMATICAL proofs , *SET theory , *ALGORITHMS , *STOCHASTIC convergence , *PROBLEM solving - Abstract
The generalized Nash equilibriumproblem(GNEP) is anextensionof the standard Nash equilibrium problem(NEP), inwhich each player's strategy set may depend on the rival player's strategies. In this paper, we present two descent type methods. The algorithms are based on a reformulation of the generalized Nash equilibrium using Nikaido-Isoda function as unconstrained optimization. We prove that our algorithms are globally convergent and the convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of the GNEP. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
39. Damped Algorithms for the Split Fixed Point and Equilibrium Problems.
- Author
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Li-Jun Zhu and Minglun Ren
- Subjects
- *
ALGORITHMS , *FIXED point theory , *VARIATIONAL inequalities (Mathematics) , *PROBLEM solving , *STOCHASTIC convergence , *MATHEMATICAL analysis - Abstract
The main purpose of this paper is to study the split fixed point and equilibrium problems which includes fixed point problems, equilibrium problems, and variational inequality problems as special cases. A damped algorithm is presented for solving this split common problem. Strong convergence analysis is shown. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
40. Algorithmic Approach to the Split Problems.
- Author
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Ming Ma
- Subjects
- *
ALGORITHMS , *PROBLEM solving , *VARIATIONAL inequalities (Mathematics) , *EQUILIBRIUM , *STOCHASTIC convergence - Abstract
This paper deals with design algorithms for the split variational inequality and equilibriumproblems. Strong convergence theorems are demonstrated. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
41. A Newton-Type Algorithm for Solving Problems of Search Theory.
- Author
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Liping Zhang
- Subjects
NEWTON-Raphson method ,PROBLEM solving ,SEARCH theory ,ALGORITHMS ,STOCHASTIC convergence ,NUMERICAL analysis - Abstract
In the survey of the continuous nonlinear resource allocation problem, Patriksson pointed out that Newton-type algorithms have not been proposed for solving the problem of search theory in the theoretical perspective. In this paper, we propose a Newton-type algorithm to solve the problem. We prove that the proposed algorithm has global and superlinear convergence. Some numerical results indicate that the proposed algorithm is promising. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
42. Structural topology optimization using an enhanced level set method.
- Author
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Shojaee, S. and Mohammadian, M.
- Subjects
HAMILTON-Jacobi equations ,MATHEMATICAL optimization ,ALGORITHMS ,LEVEL set methods ,PROBLEM solving ,NUMERICAL analysis ,STOCHASTIC convergence - Abstract
Abstract: This paper proposes an effective algorithm based on the Level Set Method (LSM) to solve the problem of topology optimization. The Hamilton–Jacobi Partial Differential Equation (H-J PDE), level set equation, is modified to increase the performance. We combine the topological derivative with nonlinear LSM to create a remedy against premature convergence and strong dependency of the optimal topology on the initial design. The magnitude of the gradient in the LS equation was replaced by several Delta functions and the results were explored. Instead of the explicit scheme, which is commonly used in conventional LSM, a semi-implicit additive operator splitting scheme was carried out in our study to solve the LS equation. A truncation strategy was implemented to limit maximum and minimum values in the design domain. Finally, several numerical examples were provided to confirm the validity of the method and show its accuracy, as well as convergence speed. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
43. On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions.
- Author
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Bingzhuang Liu, Changyu Wang, and Wenling Zhao
- Subjects
STOCHASTIC convergence ,ALGORITHMS ,PROBLEM solving ,MATHEMATICAL optimization ,ITERATIVE methods (Mathematics) ,GENERALIZATION ,MATHEMATICAL sequences - Abstract
We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian- Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
44. Convergence of a Proximal Point Algorithm for Solving Minimization Problems.
- Author
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Hamdi, Abdelouahed, Noor, M. A., and Mukheimer, A. A.
- Subjects
STOCHASTIC convergence ,ALGORITHMS ,PROBLEM solving ,SUBSTITUTIONS (Mathematics) ,CONVEX functions ,ITERATIVE methods (Mathematics) ,MATHEMATICAL proofs - Abstract
We introduce and consider a proximal point algorithm for solving minimization problems using the technique of Güler. This proximal point algorithm is obtained by substituting the usual quadratic proximal term by a class of convex nonquadratic distance-like functions. It can be seen as an extragradient iterative scheme. We prove the convergence rate of this new proximal point method under mild assumptions. Furthermore, it is shown that this estimate rate is better than the available ones. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
45. Convergence of a FEM and two-grid algorithms for elliptic problems on disjoint domains
- Author
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Jovanovic, Boško S., Koleva, Miglena N., and Vulkov, Lubin G.
- Subjects
- *
STOCHASTIC convergence , *FINITE element method , *ALGORITHMS , *MATHEMATICAL decoupling , *NUMERICAL analysis , *RECTANGLES , *PROBLEM solving - Abstract
Abstract: In this paper, we analyze a FEM and two-grid FEM decoupling algorithms for elliptic problems on disjoint domains. First, we study the rate of convergence of the FEM and, in particular, we obtain a superconvergence result. Then with proposed algorithms, the solution of the multi-component domain problem (simple example — two disjoint rectangles) on a fine grid is reduced to the solution of the original problem on a much coarser grid together with solution of several problems (each on a single-component domain) on fine meshes. The advantage is the computational cost although the resulting solution still achieves asymptotically optimal accuracy. Numerical experiments demonstrate the efficiency of the algorithms. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
46. An augmented Lagrangian fish swarm based method for global optimization
- Author
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Rocha, Ana Maria A.C., Martins, Tiago F.M.C., and Fernandes, Edite M.G.P.
- Subjects
- *
LAGRANGE equations , *MATHEMATICAL optimization , *STOCHASTIC analysis , *ALGORITHMS , *CONSTRAINED optimization , *PROBLEM solving , *STOCHASTIC convergence , *MATHEMATICAL functions - Abstract
Abstract: This paper presents an augmented Lagrangian methodology with a stochastic population based algorithm for solving nonlinear constrained global optimization problems. The method approximately solves a sequence of simple bound global optimization subproblems using a fish swarm intelligent algorithm. A stochastic convergence analysis of the fish swarm iterative process is included. Numerical results with a benchmark set of problems are shown, including a comparison with other stochastic-type algorithms. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
47. Projection algorithms for the system of mixed variational inequalities in Banach spaces
- Author
-
Zhang, Qing-bang, Deng, Ruliang, and Liu, Liu
- Subjects
- *
VARIATIONAL inequalities (Mathematics) , *BANACH spaces , *ALGORITHMS , *PROBLEM solving , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence - Abstract
Abstract: In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
48. Robust Optimal Power Flow Solution Using Trust Region and Interior-Point Methods.
- Author
-
Sousa, Andréa A., Torres, Geraldo L., and Canizares, Claudio A.
- Subjects
ELECTRIC power systems ,ELECTRICAL load ,MATHEMATICAL optimization ,ALGORITHMS ,PROBLEM solving ,STOCHASTIC convergence ,APPROXIMATION theory ,ROBUST control - Abstract
A globally convergent optimization algorithm for solving large nonlinear optimal power flow (OPF) problems is presented. As power systems become heavily loaded, there is an increasing need for globally convergent OPF algorithms. By global convergence, one means the optimization algorithm being able to converge to an OPF solution, if at least one exists, for any choice of initial point. The globally convergent OPF presented is based on an infinity-norm trust region approach, using interior-point methods to solve the trust region subproblems. The performance of the proposed trust region interior-point OPF algorithm, when applied to the IEEE 30-, 57-, 118-, and 300-bus systems, and to an actual 1211-bus system, is compared with that of two widely used nonlinear interior-point methods, namely, a pure primal-dual and its predictor-corrector variant. [ABSTRACT FROM PUBLISHER]
- Published
- 2011
- Full Text
- View/download PDF
49. IGAOR and multisplitting IGAOR methods for linear complementarity problems
- Author
-
Li, Sheng-Guo, Jiang, Hao, Cheng, Li-Zhi, and Liao, Xiang-Ke
- Subjects
- *
STOCHASTIC convergence , *LINEAR complementarity problem , *NUMERICAL analysis , *MATHEMATICAL analysis , *ALGORITHMS , *PROBLEM solving - Abstract
Abstract: In this paper, we propose an interval version of the generalized accelerated overrelaxation methods, which we refer to as IGAOR, for solving the linear complementarity problems, LCP (M, q), and develop a class of multisplitting IGAOR methods which can be easily implemented in parallel. In addition, in regards to the H-matrix with positive diagonal elements, we prove the convergence of these algorithms and illustrate their efficiency through our numerical results. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
50. A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces
- Author
-
Wang, Ziming, Su, Yongfu, Wang, Dongxing, and Dong, Yucai
- Subjects
- *
ITERATIVE methods (Mathematics) , *NONEXPANSIVE mappings , *PROBLEM solving , *BANACH spaces , *STOCHASTIC convergence , *ALGORITHMS - Abstract
Abstract: In this paper, we prove strong convergence theorems by the hybrid method for a family of hemi-relatively nonexpansive mappings in a Banach space. Our results improve and extend the corresponding results given by Qin et al. [Xiaolong Qin, Yeol Je Cho, Shin Min Kang, Haiyun Zhou, Convergence of a modified Halpern-type iteration algorithm for quasi--nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051–1055], and at the same time, our iteration algorithm is different from the Kimura and Takahashi algorithm, which is a modified Mann-type iteration algorithm [Yasunori Kimura, Wataru Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in Banach space, J. Math. Anal. Appl. 357 (2009) 356–363]. In addition, we succeed in applying our algorithm to systems of equilibrium problems which contain a family of equilibrium problems. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
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