Lagrangian variational inequalities feature both primal and dual elements in expressing first-order conditions for optimality in a wide variety of settings where "multipliers" in a very general sense need to...
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Lagrangian variational inequalities feature both primal and dual elements in expressing first-order conditions for optimality in a wide variety of settings where "multipliers" in a very general sense need to be brought in. Their stochastic version relates to problems of stochastic programming and covers not only classical formats with inequality constraints but also composite models with nonsmooth objectives. The progressive hedging algorithm, as a means of solving stochastic programming problems, has however focused so far only on optimality conditions that correspond to variational inequalities in primal variables alone. Here that limitation is removed by appealing to a recent extension of progressive hedging to multistage stochastic variational inequalities in general.
A class of nonlinear operators in Banach spaces is proposed. We call each operator in this class a firmly nonexpansive-type mapping. This class contains the classes of firmly nonexpansive mappings in Hilbert spaces an...
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A class of nonlinear operators in Banach spaces is proposed. We call each operator in this class a firmly nonexpansive-type mapping. This class contains the classes of firmly nonexpansive mappings in Hilbert spaces and resolvents of maximal monotone operators in Banach spaces. We study the existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.
We propose a new modified proximal point algorithm combined with Halpern's iteration process for nonexpansive mappings in the framework of CAT(0) spaces. We establish a strong convergence theorem under some mild c...
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We propose a new modified proximal point algorithm combined with Halpern's iteration process for nonexpansive mappings in the framework of CAT(0) spaces. We establish a strong convergence theorem under some mild conditions. Our results extend some known results which appeared in the literature.
作者:
Sipos, AndreiUniv Bucharest
Dept Comp Sci Fac Math & Comp Sci Res Ctr Log Optimizat & Secur LOS Bucharest Romania Romanian Acad
Simion Stoilow Inst Math Bucharest Romania
Recently, the author, together with L. Leustean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal metho...
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Recently, the author, together with L. Leustean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families.
Linearly constrained separable convex minimization problems have been raised widely in many real-world *** this paper,we propose a homotopy-based alternating direction method of multipliers for solving this kind of **...
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Linearly constrained separable convex minimization problems have been raised widely in many real-world *** this paper,we propose a homotopy-based alternating direction method of multipliers for solving this kind of *** proposed method owns some advantages of the classical proximal alternating direction method of multipliers and homotopy *** some suitable condi-tions,we prove global convergence and the worst-case O(k/1)convergence rate in a nonergodic *** numerical results indicate effectiveness and efficiency of the proposed method compared with some state-of-the-art methods.
The alternating direction method of multipliers (ADMM) is a classical effective method for solving two-block convex optimization subject to linear constraints. However, its convergence may not be guaranteed for multip...
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The alternating direction method of multipliers (ADMM) is a classical effective method for solving two-block convex optimization subject to linear constraints. However, its convergence may not be guaranteed for multiple-block case without additional assumptions. One remedy might be the block-wise ADMM (BADMM), in which the variables are regrouped into two groups firstly and then the augmented Lagrangian function is minimized w.r.t. each block variable by the following scheme: using a Gauss-Seidel fashion to update the variables between each group, while using a Jacobi fashion to update the variables within each group. In order to derive its convergence property, a special proximal term is added to each subproblem. In this paper, we propose a new partial PPA block-wise ADMM where we only need to add proximal terms to the subproblems in the first group. At the end of each iteration, an extension step on all variables is performed with a fixed step size. As the subproblems in the second group are unmodified, the resulting sequence might yield better quality as well as potentially faster convergence speed. Preliminary experimental results show that the new algorithm is empirically effective on solving both synthetic and real problems when it is compared with several very efficient ADMM-based algorithms.
It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge s...
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It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with non-summable error sequence is also discussed.
Let H he a real Hilbert space and let T:H --> 2(H) he a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to genera...
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Let H he a real Hilbert space and let T:H --> 2(H) he a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit v epsilon T(-1)0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration. (C) 2000 Academic Press.
In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mappi...
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In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented.
We study equilibrium problem on general Riemannian manifolds, focusing on existence of solutions and the convexity properties of the solution set. Our approach consists in relating the equilibrium problem to a suitabl...
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We study equilibrium problem on general Riemannian manifolds, focusing on existence of solutions and the convexity properties of the solution set. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. We apply our results to the mixed variational inequality and the Nash equilibrium problem. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388 (2012) 61-77 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds. (C) 2019 Elsevier Inc. All rights reserved.
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