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A WEAKTOSTRONGCONVERGENCE PRINCIPLE FOR FEJÉRMONOTONE METHODS IN HILBERT SPACES
, 2001
"... We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assump ..."
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Cited by 36 (8 self)
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We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assumptions. Several applications are discussed.
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
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Learning Dynamics In Mechanism Design: An Experimental Comparison Of Public Goods Mechanisms
, 2003
"... In a repeatedinteraction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in oneshot games. Inspired by results obtained in previous experiments, the current paper proposes a simple best response model in which players' beliefs are func ..."
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Cited by 10 (2 self)
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In a repeatedinteraction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in oneshot games. Inspired by results obtained in previous experiments, the current paper proposes a simple best response model in which players' beliefs are functions of previous strategy profiles. The predictions of the model are found to be highly consistent with new experimental data from five mechanisms with various types of equilibria. Interesting properties of a 2parameter VickreyClarkeGroves mechanism help to draw out this result. The simplicity of the model makes it useful in predicting dynamic stability of other mechanisms.
Generalized Mann iterates for constructing fixed points in Hilbert spaces
"... this paper is to introduce and analyze a common algorithmic framework encompassing and extending the above iterative methods. The algorithm under consideration is the following inexact, Mannlike generalization of (5) xn+1 = xn + n Tnxn + e n xn where e n 2 H; 0 < n < 2; and Tn 2 T : (10) Here, ..."
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Cited by 2 (0 self)
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this paper is to introduce and analyze a common algorithmic framework encompassing and extending the above iterative methods. The algorithm under consideration is the following inexact, Mannlike generalization of (5) xn+1 = xn + n Tnxn + e n xn where e n 2 H; 0 < n < 2; and Tn 2 T : (10) Here, e n stands for the error made in the computation of Tnxn ; incorporating such errors provides a more realistic model of the actual implementation of the algorithm. Throughout, the convex combinations in (10) are de ned as xn = n;j x j ; (11) 3 where ( n;j ) n;j0 are the entries of an in nite lower triangular row stochastic matrix A, i.e., > > (8j 2 N) n;j 0 (8j 2 N) j > n ) n;j = 0 j=0 n;j = 1; (12) which satis es the regularity condition (8j 2 N) lim n!+1 n;j = 0: (13) Our analysis will not rely on the segmenting condition (7) and will provide convergence results for the inexact, extended Mann iterations (10) for a wide range of averaging schemes
Common Fixed Point Iterations of a Finite Family of Quasinonexpansive Maps
"... Abstract—It is proved that Kuhfittig iteration process converges to a common fixed point of a finite family of quasinonexpansive maps on a Banach space. This result is extended to the random case. Our work improves upon several wellknown results in the current literature. Keywords: Quasinonexpans ..."
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Abstract—It is proved that Kuhfittig iteration process converges to a common fixed point of a finite family of quasinonexpansive maps on a Banach space. This result is extended to the random case. Our work improves upon several wellknown results in the current literature. Keywords: Quasinonexpansive map, common fixed point, iteration process, Banach space, measurable space 1
APPROXIMATING FIXED POINTS OF NONEXPANSIVE MAPPINGS BY
"... Let D be a subset of a normed space X and T: D → X be a nonexpansive mapping. In this paper we consider the following iteration method which generalizes Ishikawa iteration process: xn+1 = t (1) n T (t (2) n T ( · · · T (t (k) n T xn + (1 − t (k) n)xn + u (k) n) + · · ·) +(1 − t (2) n)xn + u (2) ..."
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Let D be a subset of a normed space X and T: D → X be a nonexpansive mapping. In this paper we consider the following iteration method which generalizes Ishikawa iteration process: xn+1 = t (1) n T (t (2) n T ( · · · T (t (k) n T xn + (1 − t (k) n)xn + u (k) n) + · · ·) +(1 − t (2) n)xn + u (2) n) + (1 − t (1) n)xn + u (1) n, n = 1, 2, 3..., where 0 ≤ t (i) n ≤ 1 for all n ≥ 1 and i = 1,..., k, and sequences {xn} and {u (i) n}, i = 1,..., k, are in D. We improve several results in [2], concerning approximation of fixed points of T. 1.