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.
Due to its significant efficiency, the alternating direction method (ADM) has attracted a lot of attention in solving linearly constrained structured convex optimization. In this paper, in order to make implementation...
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Due to its significant efficiency, the alternating direction method (ADM) has attracted a lot of attention in solving linearly constrained structured convex optimization. In this paper, in order to make implementation of ADM relatively easy, some linearized proximal ADMs are proposed and the associated convergence results of the proposed linearized proximal ADMs are given. Additionally, theoretical analysis shows that the relaxation factor for the linearized proximal ADMs can have the same restriction region as that for the general ADM.
Motivated by computing medians and means, two proximal splitting methods, backward-backward scheme introduced by Passty [Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J Math Anal Ap...
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Motivated by computing medians and means, two proximal splitting methods, backward-backward scheme introduced by Passty [Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J Math Anal Appl. 1979;72:383-390.] and barycentric method introduced by Lehdili and Lemaire [Metric spaces of non-positive curvature. Berlin: Springer;1999.] are investigated in Hadamard space setting. A weak ergodic convergence theorem and some strong convergene results for both schemes are studied.
Given a Hilbert space H and a closed convex function Phi : H -> R boolean OR {+infinity}, we consider the inertial proximalalgorithm x(n+1) - x(n) - alpha(n)(x(n) - x(n-1)) + beta(n)partial derivative Phi(x(n+1)) ...
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Given a Hilbert space H and a closed convex function Phi : H -> R boolean OR {+infinity}, we consider the inertial proximalalgorithm x(n+1) - x(n) - alpha(n)(x(n) - x(n-1)) + beta(n)partial derivative Phi(x(n+1)) (sic) 0, (A) where (alpha(n)) and (beta(n)) are nonnegative sequences. The notation partial derivative Phi stands for the subdifferential of Phi in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to inf Phi as n -> +infinity. When Phi has a unique minimum, the question of the convergence of (x(n)) is solved. In the case of multiple minima, it is proved that if (Pi k=1n alpha k) is not an element of l(1) and if a suitable geometric condition on the set argmin Phi is fulfilled, then non stationary sequences of (A) cannot converge.
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.
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.
We first introduce the notion of weak sharpness for the solution sets of variational inequality problems (in short, VIP) on Hadamard spaces. We then study the finite convergence property of sequences generated by the ...
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We first introduce the notion of weak sharpness for the solution sets of variational inequality problems (in short, VIP) on Hadamard spaces. We then study the finite convergence property of sequences generated by the inexact proximal point algorithm with different error terms for solving VIP under weak sharpness of the solution set. We also give an upper bound on the number of iterations by which the sequence generated by the exact proximal point algorithm converges to a solution of VIP. An example is also given to illustrate our results.
A common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model, has been to set up and analyze a numbe...
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A common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model, has been to set up and analyze a number scenarios. The aim then is to identify trends and essential features on which a robust decision policy can be based. This paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios. The scenarios are bundled at various levels to reflect the availability of information, and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight.
In this paper, two iteration processes are used to find the solutions of the mathematical programming for the sum of two convex functions. In infinite Hilbert space, we establish two strong convergence theorems as reg...
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In this paper, two iteration processes are used to find the solutions of the mathematical programming for the sum of two convex functions. In infinite Hilbert space, we establish two strong convergence theorems as regards this problem. As applications of our results, we give strong convergence theorems as regards the split feasibility problem with modified CQ method, strong convergence theorem as regards the lasso problem, and strong convergence theorems for the mathematical programming with a modified proximal point algorithm and a modified gradient-projection method in the infinite dimensional Hilbert space. We also apply our result on the lasso problem to the image deblurring problem. Some numerical examples are given to demonstrate our results. The main result of this paper entails a unified study of many types of optimization problems. Our algorithms to solve these problems are different from any results in the literature. Some results of this paper are original and some results of this paper improve, extend, and unify comparable results in existence in the literature.
The Bregman function-based proximal point algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additiona...
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The Bregman function-based proximal point algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) proximal point algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding - for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned above. Hence, though still of interior proximal type, the suggested method is applicable to an essentially larger class of equilibrium problems, especially including non-uniquely solvable ones.
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