The Experts below are selected from a list of 889965 Experts worldwide ranked by ideXlab platform
Hideaki Iiduka - One of the best experts on this subject based on the ideXlab platform.
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Acceleration method for convex optimization over the fixed point Set of a nonexpansive mapping
Mathematical Programming, 2014Co-Authors: Hideaki IidukaAbstract:The existing algorithms for solving the convex minimization problem over the fixed point Set of a nonexpansive mapping on a Hilbert space are based on algorithmic methods, such as the steepest descent method and conjugate gradient methods, for finding a minimizer of the objective function over the whole space, and attach importance to minimizing the objective function as quickly as possible. Meanwhile, it is of practical importance to devise algorithms which converge in the fixed point Set quickly because the fixed point Set is the Set with the constraint conditions that must be satisfied in the problem. This paper proposes an algorithm which not only minimizes the objective function quickly but also converges in the fixed point Set much faster than the existing algorithms and proves that the algorithm with diminishing step-size sequences strongly converges to the solution to the convex minimization problem. We also analyze the proposed algorithm with each of the Fletcher---Reeves, Polak---Ribiere---Polyak, Hestenes---Stiefel, and Dai---Yuan formulas used in the conventional conjugate gradient methods, and show that there is an inconvenient possibility that their algorithms may not converge to the solution to the convex minimization problem. We numerically compare the proposed algorithm with the existing algorithms and show its effectiveness and fast convergence.
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Approximate solutions to variational inequality over the fixed point Set of a strongly nonexpansive mapping
Fixed Point Theory and Applications, 2014Co-Authors: Shigeru Iemoto, Kazuhiro Hishinuma, Hideaki IidukaAbstract:Variational inequality problems over fixed point Sets of nonexpansive mappings include many practical problems in engineering and applied mathematics, and a number of iterative methods have been presented to solve them. In this paper, we discuss a variational inequality problem for a monotone, hemicontinuous operator over the fixed point Set of a strongly nonexpansive mapping on a real Hilbert space. We then present an iterative algorithm, which uses the strongly nonexpansive mapping at each iteration, for solving the problem. We show that the algorithm potentially converges in the fixed point Set faster than algorithms using firmly nonexpansive mappings. We also prove that, under certain assumptions, the algorithm with slowly diminishing step-size sequences converges to a solution to the problem in the sense of the weak topology of a Hilbert space. Numerical results demonstrate that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm.
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fixed point optimization algorithm and its application to network bandwidth allocation
Journal of Computational and Applied Mathematics, 2012Co-Authors: Hideaki IidukaAbstract:A convex optimization problem for a strictly convex objective function over the fixed point Set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.
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three term conjugate gradient method for the convex optimization problem over the fixed point Set of a nonexpansive mapping
Applied Mathematics and Computation, 2011Co-Authors: Hideaki IidukaAbstract:Many constrained Sets in problems such as signal processing and optimal control can be represented as a fixed point Set of a certain nonexpansive mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point Set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm.
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A new iterative algorithm for the variational inequality problem over the fixed point Set of a firmly nonexpansive mapping
Optimization, 2010Co-Authors: Hideaki IidukaAbstract:Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point Set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point Set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point Set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point Set are provided in the final section.
Rafal Zalas - One of the best experts on this subject based on the ideXlab platform.
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an algorithm for solving the variational inequality problem over the fixed point Set of a quasi nonexpansive operator in euclidean space
arXiv: Optimization and Control, 2013Co-Authors: Andrzej Cegielski, Aviv Gibali, Simeon Reich, Rafal ZalasAbstract:This paper is concerned with the variational inequality problem (VIP) over the fixed point Set of a quasi-nonexpansive operator. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem.
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an algorithm for solving the variational inequality problem over the fixed point Set of a quasi nonexpansive operator in euclidean space
Numerical Functional Analysis and Optimization, 2013Co-Authors: Andrzej Cegielski, Aviv Gibali, Simeon Reich, Rafal ZalasAbstract:This article is concerned with the variational inequality problem VIP(ℱ, Fix(T)): find such that for all z ∈ Fix(T), where T: ℝ n → ℝ n is quasi-nonexpansive, Fix(T) is its nonempty fixed point Set, and ℱ: ℝ n → ℝ n is monotone. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem.
Douglas M Pickrell - One of the best experts on this subject based on the ideXlab platform.
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the diagonal distribution for the invariant measure of a unitary type symmetric space
Transformation Groups, 2006Co-Authors: Douglas M PickrellAbstract:Let Θ denote an involution for a simply connected compact Lie group U, let K denote the fixed point Set, and let μ denote the U-invariant probability measure on U/K. Consider the geodesic embedding $\phi :U/K\to U:u\mapsto uu^{ -\Theta}$ of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for $\phi_{*}\mu,$ relative to a compatible triangular decomposition of G, the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space G0/K discovered by Evens and Lu.
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the diagonal distribution for the invariant measure of a unitary type symmetric space
arXiv: Symplectic Geometry, 2005Co-Authors: Douglas M PickrellAbstract:Let U denote a simply connected compact Lie group, let K denote the fixed point Set for an involutive automorphism of U, and let m denote the U-invariant probability measure on the symmetric space U/K. Consider the geodesic embedding U/K into U of Cartan. In this paper we compute the diagonal distribution for m, relative to a compatible triangular decomposition of G, the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space G_0/K discovered by Evens and Lu.
Aleksandra Borówka - One of the best experts on this subject based on the ideXlab platform.
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Quaternion-Kähler manifolds near maximal fixed point Sets of $$S^1$$ S 1
Annali di Matematica Pura ed Applicata (1923 -), 2019Co-Authors: Aleksandra BorówkaAbstract:Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating $$S^1$$ S 1 -symmetry with the fixed point Set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating $$S^1$$ S 1 -symmetry induce on the fixed point Set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point Set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.
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C-projective symmetries of submanifolds in quaternionic geometry
Annals of Global Analysis and Geometry, 2019Co-Authors: Aleksandra Borówka, Henrik WintherAbstract:The generalized Feix–Kaledin construction shows that c-projective 2 n -manifolds with curvature of type (1, 1) are precisely the submanifolds of quaternionic 4 n -manifolds which are Fixed-Point Set of a special type of quaternionic circle action. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type (1, 1) curvature is a submanifold of a submaximally symmetric quaternionic model and show how this fits into the construction. We give conditions for when the c-projective symmetries extend from the Fixed-Point Set of the circle action to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi and Eguchi–Hanson hyperkähler structures, showing that in some cases all quaternionic symmetries are obtained in this way.
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C-projective symmetries of submanifolds in quaternionic geometry
Annals of Global Analysis and Geometry, 2019Co-Authors: Aleksandra Borówka, Henrik WintherAbstract:The generalized Feix–Kaledin construction shows that c-projective 2 n -manifolds with curvature of type (1, 1) are precisely the submanifolds of quaternionic 4 n -manifolds which are Fixed-Point Set of a special type of quaternionic circle action. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type (1, 1) curvature is a submanifold of a submaximally symmetric quaternionic model and show how this fits into the construction. We give conditions for when the c-projective symmetries extend from the Fixed-Point Set of the circle action to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi and Eguchi–Hanson hyperkähler structures, showing that in some cases all quaternionic symmetries are obtained in this way.
Eric L. Mcdowell - One of the best experts on this subject based on the ideXlab platform.
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Fixed point Set characterizations of Peano continua and absolute retracts
Topology and its Applications, 2003Co-Authors: Eric L. McdowellAbstract:Abstract We extend the notion of absolute fixed point Sets to the Setting of continuum-valued maps whose point images have small diameters. We demonstrate that the resulting class of spaces ( e -MAFS) coincides with the class of absolute fixed point Sets and with the class of absolute retracts in the one-dimensional and planar Settings, but that the class of n -dimensional absolute fixed point Sets is strictly contained in e -MAFS for all n >1. Moreover, we will give two characterizations for the class of Peano continua in terms of e -MAFS.
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Absolute fixed point Sets for multi-valued maps
Proceedings of the American Mathematical Society, 1998Co-Authors: Eric L. McdowellAbstract:The notion of a multi-valued absolute fixed point Set (MAFS) will be defined and characterized in the Setting of Set-valued maps with images containing multiple components.
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Absolute fixed point Sets for continuum-valued maps
Proceedings of the American Mathematical Society, 1996Co-Authors: Eric L. Mcdowell, Sam B. NadlerAbstract:The notion of an absolute fixed point Set in the Setting of continuumvalued maps will be defined and characterized.
