Let H be a Hilbert space and A, B: H paired right arrows H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm: (x(n+2)-2x(n+1)+x(n))/lambda(2)(n)+A((x...
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Let H be a Hilbert space and A, B: H paired right arrows H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm: (x(n+2)-2x(n+1)+x(n))/lambda(2)(n)+A((x(n+2)-x(n+1))/lambda(n))+B(x(n+2))there exists 0, (A) where (lambda(n)) is a sequence of positive steps. algorithm (A) may be viewed as the discretized equation of a nonlinear oscillator subject to friction. We prove that, if 0 is an element of e int (A) is strongly convergent and its limit x(infinity) satisfies 0 is an element of A(0)+B(x(infinity)). We show that, under a general condition, the limit x(infinity) is achieved in a finite number of iterations. When this condition is not satisfied, we prove in a rather large setting that the convergence rate is at least geometrical.
Let C be a nonempty closed convex subset of a real Hilbert space and let {T-n} be a family of mappings of C into itself such that the set of all common fixed points of {T-n} is nonempty. We consider a sequence {x(n)} ...
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Let C be a nonempty closed convex subset of a real Hilbert space and let {T-n} be a family of mappings of C into itself such that the set of all common fixed points of {T-n} is nonempty. We consider a sequence {x(n)} generated by the hybrid method in mathematical programming and give the conditions of {T-n} under which {x(n)} converges strongly to a common fixed point of {T-n}.
In this paper, we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem (VIP), where the finite convergence means that the algorithm can find an exact...
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In this paper, we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem (VIP), where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations. By using this result, we obtain that the well-known proximal point algorithm possesses finite convergence if the solution set of VIP is weakly sharp. As an extension, we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g-sharpness. (c) 2005 Elsevier Inc. All rights reserved.
In 1991, Guler constructed a proximalpoint iteration that converges weakly but not in norm. By building on a recent result of Hundal, we present a new, considerably simpler, example of this type.
In 1991, Guler constructed a proximalpoint iteration that converges weakly but not in norm. By building on a recent result of Hundal, we present a new, considerably simpler, example of this type.
This paper studies the convergence of the classical proximal point algorithm without assuming monotonicity of the underlying mapping. Practical conditions are given that guarantee the local convergence of the iterates...
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This paper studies the convergence of the classical proximal point algorithm without assuming monotonicity of the underlying mapping. Practical conditions are given that guarantee the local convergence of the iterates to a solution of T(x) B 0, where T is an arbitrary set-valued mapping from a Hilbert space to itself. In particular, when the problem is "nonsingular" in the sense that T-1 has a Lipschitz localization around one of the solutions, local linear convergence is obtained. This kind of regularity property of variational inclusions has been extensively studied in the literature under the name of strong regularity, and it can be viewed as a natural generalization of classical constraint qualifications in nonlinear programming to more general problem classes. Combining the new convergence results with an abstract duality framework for variational inclusions, the author proves the local convergence of multiplier methods for a very general class of problems. This gives as special cases new convergence results for multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming.
Recently, Hundal has constructed a hyperplane H, a cone K, and a starting point y(0) in l(2) such that the sequence of alternating projections ((PKPH)(n)y(0))nepsilonN converges weakly to some point in H boolean AND K...
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Recently, Hundal has constructed a hyperplane H, a cone K, and a starting point y(0) in l(2) such that the sequence of alternating projections ((PKPH)(n)y(0))nepsilonN converges weakly to some point in H boolean AND K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections;hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (a la Genel and Lindenstrauss) and for the proximal point algorithm (a la Guler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor et at. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well. (C) 2003 Elsevier Ltd. All rights reserved.
Optimiz at ion models are sometimes promoted because they provide "optimal" solutions as defined by a cost model. Simulation models, by contrast, are guided by rules that are specified by experts in operatio...
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Optimiz at ion models are sometimes promoted because they provide "optimal" solutions as defined by a cost model. Simulation models, by contrast, are guided by rules that are specified by experts in operations. While these may seem heuristic in nature, they often reflect issues that are difficult to capture in a cost-based objective function. "Optimizing simulators" combine the intelligence of optimization with the flexibility of simulation in the handling of system dynamics, but still suffer from the limitation that the behavior is entirely determined by a cost model. In this paper, we show how a cost-based model can be guided through a set of low-dimensional patterns which are essentially simple rules determined by a domain expert. Patterns are incorporated through a penalty term, scaled by a coefficient that controls that tradeoff between minimizing costs and minimizing the difference between model behavior and the exogenous patterns. (C) 2004 Elsevier Ltd. All rights reserved.
In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of general...
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In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877-898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226-2401. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.
A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves ...
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A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as for various splitting methods for finding a zero of the sum of monotone operators.
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