Point Sets Of Nonexpansive Mappings Research Articles

Decentralized hierarchical constrained convex optimization

This paper proposes a decentralized optimization algorithm for the triple-hierarchical constrained convex optimization problem of minimizing a sum of strongly convex functions subject to a paramonotone variational inequality constraint over an intersection of fixed point sets of nonexpansive mappings. The existing algorithms for solving this problem are centralized optimization algorithms using all the information in the problem, and these algorithms are effective, but only under certain additional restrictions. The main contribution of this paper is to present a convergence analysis of the proposed algorithm in order to show that the proposed algorithm using incremental gradients with diminishing step-size sequences converges to the solution to the problem without any additional restrictions. Another contribution of this paper is the elucidation of the practical applications of hierarchical constrained optimization in the form of network resource allocation and optimal control problems. In particular, it is shown that the proposed algorithm can be applied to decentralized network resource allocation with a triple-hierarchical structure.

Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.

Two stochastic optimization algorithms for convex optimization with fixed point constraints

Two optimization algorithms are proposed for solving a stochastic programming problem for which the objective function is given in the form of the expectation of convex functions and the constraint set is defined by the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. This setting of fixed point constraints enables consideration of the case in which the projection onto each of the constraint sets cannot be computed efficiently. Both algorithms use a convex function and a nonexpansive mapping determined by a certain probabilistic process at each iteration. One algorithm blends a stochastic gradient method with the Halpern fixed point algorithm. The other is based on a stochastic proximal point algorithm and the Halpern fixed point algorithm; it can be applied to nonsmooth convex optimization. Convergence analysis showed that, under certain assumptions, any weak sequential cluster point of the sequence generated by either algorithm almost surely belongs to the solution set of the problem. Convergence rate analysis illustrated their efficiency, and the numerical results of convex optimization over fixed point sets demonstrated their effectiveness.

Novel Hybrid Methods for Pseudomonotone Equilibrium Problems and Common Fixed Point Problems

ABSTRACTThe purpose of this article is to introduce some hybrid algorithms for finding a common element of the solution sets of pseudomonotone equilibrium problems and the fixed point sets of nonexpansive mappings in real Hilbert spaces. Our algorithms combine Mann’s iterative methods and Armijo line-search with parallel splitting-up and hybrid techniques. The strong convergence of the proposed algorithms are established without the assumption on the Lipschitz-type condition for the bifunctions involved.

Incremental subgradient method for nonsmooth convex optimization with fixed point constraints

This paper proposes an incremental subgradient method for solving the problem of minimizing the sum of nondifferentiable, convex objective functions over the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. The proposed algorithm can work in nonsmooth optimization over constraint sets onto which projections cannot be always implemented, whereas the conventional incremental subgradient method can be applied only when a constraint set is simple in the sense that the projection onto it can be easily implemented. We first study its convergence for a constant step size. The analysis indicates that there is a possibility that the algorithm with a small constant step size approximates a solution to the problem. Next, we study its convergence for a diminishing step size and show that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. Moreover, we show the whole sequence generated by the algorithm with a diminishing step size strongly converges to the solution to the problem under certain assumptions. We also give examples of real applied problems which satisfy the assumptions in the convergence theorems and numerical examples to support the convergence analyses.

Proximal point algorithms for nonsmooth convex optimization with fixed point constraints

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algorithm for nonsmooth convex optimization, with a Halpern-type fixed point iteration algorithm. The other is based on an incremental subgradient method and the Krasnosel’skiĭ–Mann fixed point algorithm. It is shown that any weak sequential cluster point of the sequence generated by the Halpern-type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the Krasnosel’skiĭ–Mann-type algorithm, which also belongs to the solution set. Numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence.

Parallel computing subgradient method for nonsmooth convex optimization over the intersection of fixed point sets of nonexpansive mappings

Nonsmooth convex optimization problems are solved over fixed point sets of nonexpansive mappings by using a distributed optimization technique. This is done for a networked system with an operator, who manages the system, and a finite number of users, by solving the problem of minimizing the sum of the operator’s and users’ nondifferentiable, convex objective functions over the intersection of the operator’s and users’ convex constraint sets in a real Hilbert space. We assume that each of their constraint sets can be expressed as the fixed point set of an implementable nonexpansive mapping. This setting allows us to discuss nonsmooth convex optimization problems in which the metric projection onto the constraint set cannot be calculated explicitly. We propose a parallel subgradient algorithm for solving the problem by using the operator’s attribution such that it can communicate with all users. The proposed algorithm does not use any proximity operators, in contrast to conventional parallel algorithms for nonsmooth convex optimization. We first study its convergence property for a constant step-size rule. The analysis indicates that the proposed algorithm with a small constant step size approximates a solution to the problem. We next consider the case of a diminishing step-size sequence and prove that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. We also give numerical examples to support the convergence analyses.

FROM EQUILIBRIUM PROBLEMS AND FIXED POINTS PROBLEMS TO MINIMIZATION PROBLEMS

Algorithms approach to equilibrium problems and fixed points problems have been extensively studied in the literature. The purpose of this paper is devoted to consider the minimization problem of finding a point $x^†$ with the property \begin{equation*} x^†\in \Omega\quad {\rm and}\quad \|x^†\|^2=\min_{x\in \Omega}\|x\|^2, \end{equation*} where $\Omega$ is the intersection of the solution set of equilibrium problem and the fixed points set of nonexpansive mapping. For this purpose, we suggest two algorithms: \begin{eqnarray*} F(z_t,y)+\frac{1}{\lambda}\Big{\langle} y-z_t,z_t-\Big{(}(1-t)I-\lambda A\Big{)}Sz_t\Big{\rangle}\ge 0,\;\forall y\in C. \end{eqnarray*} and \begin{eqnarray*} \begin{cases} F(z_n,y)+\langle Ax_n,y-z_n\rangle+\frac{1}{\lambda_n}\Big{\langle}y-z_n,z_n-(1-\alpha_n)x_n\Big{\rangle}\ge 0,\;\forall y\in C,\\ x_{n+1}=\beta_nx_n+(1-\beta_n)Sz_n,\; n\ge 0. \end{cases} \end{eqnarray*} It is shown that under some mild conditions, the net $\{z_t\}$ and the sequences $\{z_n\}$ and $\{x_n\}$ converge strongly to $\tilde{x}$ which is the unique solution of the above minimization problem. It should be point out that our suggested algorithms solve the above minimization problem without involving projection.

Approximate solutions to variational inequality over the fixed point set of a strongly nonexpansive mapping

Abstract 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. MSC:47H06, 47J20, 47J25.

Hybrid Viscosity-like Approximation Methods for General Monotone Variational Inequalities

In this paper, we introduce two implicit and explicit hybrid viscositylike approximation methods for solving a general monotone variational inequality, which covers their monotone variational inequality with $C = H$ as a special case. We use the contractions to regularize the general monotone variational inequality, where the monotone operators are the generalized complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such general monotone variational inequality includes some monotone inclusions and some convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit hybrid viscosity-like approximation methods are shown to be strongly convergent. In the meantime, these results are applied to deriving the strong convergence theorems for a general monotone variational inequality with minimization constraint. An application in hierarchical minimization is also included.

Hybrid Viscosity-like Approximation Methods for General Monotone Variational Inequalities

In this paper, we introduce two implicit and explicit hybrid viscositylike approximation methods for solving a general monotone variational inequality, which covers their monotone variational inequality with C = H as a special case. We use the contractions to regularize the general monotone variational inequality, where the monotone operators are the generalized complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such general monotone variational inequality includes some monotone inclusions and some convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit hybrid viscosity-like approximation methods are shown to be strongly convergent. In the meantime, these results are applied to deriving the strong convergence theorems for a general monotone variational inequality with minimization constraint. An application in hierarchical minimization is also included.

Explicit Hierarchical Fixed Point Approach to Variational Inequalities

An explicit hierarchical fixed point algorithm is introduced to solve monotone variational inequalities, which are governed by a pair of nonexpansive mappings, one of which is used to define the governing operator and the other to define the feasible set. These kinds of variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Strong convergence of the algorithm is proved under different circumstances of parameter selections. Applications in hierarchical minimization problems are also included.

VISCOSITY METHOD FOR HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES

A viscosity method for a hierarchical fixed point approach to variational inequality problems is presented. This method is used to solve variational inequalities where the involving operators are complements of nonexpansive mappigs and the solutions are sought in the set of the fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings.

Hybrid methods for a class of monotone variational inequalities

We use contractions to regularize a class of monotone variational inequalities, where the monotone operators are complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit schemes are shown to be strongly convergent. An application in hierarchical minimization is included.

Parallel algorithms for variational inequalities over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings

This paper presents a framework of iterative algorithms for the variational inequality problem over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings in real Hilbert spaces. Strong convergence theorems are established under a certain contraction assumption with respect to the weighted maximum norm. The proposed framework produces as a simplest example the hybrid steepest descent method, which has been developed for solving the monotone variational inequality problem over the intersection of the fixed point sets of nonexpansive mappings. An application to a generalized power control problem and numerical examples are demonstrated.

Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings

The hybrid steepest descent method is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to broad range of convexly constrained nonlinear inverse problems in real Hilbert space. In this paper, we show that the strong convergence theorem [Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications. Elsevier, pp. 473–504] of the method for nonexpansive mapping can be extended to a strong convergence theorem of the method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings called quasi-shrinking mapping. We also present a convergence theorem of the method for paramonotone variational inequality problem over the bounded fixed point set of quasi-shrinking mapping. By these generalizations, we can approximate successively to the solution of the convex optimization problem over the fixed point set of wide range of subgradient projection operators in real Hilbert space.

Iterative Approaches to Convex Minimization Problems

The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(1):165–190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33–56] in which they consider the minimization of some function θ over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θover F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition on the parameters by the more general condition lim n→∞λ n /λ n+N = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of θ over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters.

Nonstrictly Convex Minimization over the Bounded Fixed Point Set of a Nonexpansive Mapping

In this paper, we consider, in a finite dimensional real Hilbert space , the variational inequality problem VIP: find , where is nonexpansive mapping with bounded and is paramonotone and Lipschitzian over . The nonstrictly convex minimization over the bounded fixed point set of a nonexpansive mapping is a typical example of such a variational inequality problem. We show that the hybrid steepest descent method, of which convergence properties were examined in some cases for example (Yamada, I. (2000). Convex projection algorithm from POCS to Hybrid steepest descent method. The Journal of the IEICE (in Japanese) 83:616–623; Yamada, I. (2001). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S., eds. Inherently Parallel Algorithm for Feasibility and Optimization. Elsevier; Ogura, N., Yamada, I. (2002). Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23:113–137), is still applicable to the case where and T satisfy the above conditions.

Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings

Let be nonexpansive mappings on a Hilbert space H, and let be a function which has a uniformly strongly positive and uniformly bounded second (Fréchet) derivative over the convex hull of Ti(H) for some i. We first prove that Θ has a unique minimum over the intersection of the fixed point sets of all the Ti’s at some point u*. Then a cyclic hybrid steepest descent algorithm is proposed and we prove that it converges to u*. This generalizes some recent results of Wittmann (1992), Combettes (1995), Bauschke (1996), and Yamada, Ogura, Yamashita, and Sakaniwa (1997). In particular, the minimization of Θ over the intersection of closed convex sets Ci can be handled by taking Ti to be the metric projection Pci onto Ci. We also propose a modification of our algorithm to handle the inconsistent case (i.e., when is empty as well.

A note on connection properties of fixed point sets of nonexpansive mappings

A note on connection properties of fixed point sets of nonexpansive mappings