An operator splitting method for variational inequalities with partially unknown mappings

In this paper, we propose a new operator splitting method for solving a class of variational inequality problems in which part of the underlying mappings are unknown. This class of problems arises frequently from engineering, economics and transportation equilibrium problems. At each iteration, by using the information observed from the system, the method solves a system of nonlinear equations, which is well-defined. Under mild assumptions, the global convergence of the method is proved, and its efficiency is demonstrated with numerical examples. The research of D. Han is supported by NSFC grant 10501024 and NSF of Jiangsu Province at Grant No. BK2006214.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

An ADM-based splitting method for separable convex programming

We consider the convex minimization problem with linear constraints and a block-separable objective function which is represented as the sum of three functions without coupled variables. To solve this model, it is empirically effective to extend straightforwardly the alternating direction method of multipliers (ADM for short). But, the convergence of this straightforward extension of ADM is still not proved theoretically. Based on ADM’s straightforward extension, this paper presents a new splitting method for the model under consideration, which is empirically competitive to the straightforward extension of ADM and meanwhile the global convergence can be proved under standard assumptions. At each iteration, the new method corrects the output of the straightforward extension of ADM by some slight correction computation to generate a new iterate. Thus, the implementation of the new method is almost as easy as that of ADM’s straightforward extension. We show the numerical efficiency of the new method by some applications in the areas of image processing and statistics.     

An operator splitting method for nonlinear convection-diffusion equations

Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the -dimensional convection-diffusion equation . The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case , dimensional splitting is used to reduce the -dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. ReceivedFebruary 1, 1996 / Revised version received June 24, 1996     

An iterative method for finding coincidence points of two mappings

The problem of finding coincidence points of two mappings of which one is a covering, while the other satisfies the Lipschitz condition, is examined. An iterative method for finding an approximate solution to this problem is discussed. It is based on the a priori estimates derived in the paper.     

可分离凸优化问题的非精确平行分裂算法

针对一类可分离凸优化问题提出了一种非精确平行分裂算法.该算法充分利用了所求解问题的可分离结构,并对子问题进行非精确求解.在适当的条件下,证明了所提出的非精确平行分裂算法的全局收敛性,初步的数值实验说明了算法有效性.     

An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings

In this paper, an extragradient-type method is introduced for finding a common element in the solution set of generalized equilibrium problems, in the solution set of classical variational inequalities and in the fixed point set of strictly pseudocontractive mappings. It is proved that the iterative sequence generated in the purposed extragradient-type iterative process converges weakly to some common element in real Hilbert spaces.     

An optimal method for stochastic composite optimization

This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems.     

An invariance theorem for mappings

If a continuous mapping f carries the boundary of a set D into the closure of D, it may not be true that f maps D into its closure, even if f is injective on D. Examples are discussed, and conditions are given under which it is true. A simplified application is given to a biological migration–selection model.     

An E-based splitting finite element method for time-dependent eddy current equations

A new decoupled finite element method is suggested to approximate time-dependent eddy current equations in a three-dimensional polyhedral domain. This method is based on solving a vector and a scalar from the splitting of the electric field by using edge and nodal finite elements. An optimal energy-norm error estimate in finite time is obtained by introducing a projection operator.     

Continuation method for -sublinear mappings

Let be a real Banach space partially ordered by a closed convex cone with nonempty interior . We study the continuation method for the monotone operator which satisfies

for all , , where . Thompson's metric is among the main tools we are using.     

An efficient and novel numerical method for quasiconformal mappings of doubly connected domains

A numerical method for quasiconformal mapping of doubly connected domains onto annuli is presented. The ratio R of the radii of the annulus is not known a priori and is determined as part of the solution procedure. The numerical method presented in this paper requires solving iteratively a sequence of inhomogeneous Beltrami equations, each for a different R. R is updated using a procedure based on the bisection method. The new method is an extension of Daripas method for the quasiconformal mapping of the exterior of simply connected domains onto the interior of unit disks [15]. It uses fast and accurate algorithms for evaluating certain singular integrals and is, thus, very efficient and accurate. Its performance is demonstrated for several doubly connected domains.     

An intersection theorem for set-valued mappings

Given a nonempty convex set X in a locally convex Hausdorff topological vector space, a nonempty set Y and two set-valued mappings T: X ? X, S: Y ? X we prove that under suitable conditions one can find an x ∈ X which is simultaneously a fixed point for T and a common point for the family of values of S. Applying our intersection theorem, we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.     

A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings

We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

A method for analytically representing area-preserving mappings

An area-preserving mapping is considered. It is assumed that the mapping has a fixed point and is analytic in a small neighbourhood near it. A constructive algorithm for obtaining a representation of the mapping in the form of a composite of two area-preserving mappings, one of which is a nearly identity mapping, while the other corresponds to the real normal form of a linearized mapping, is described. The algorithm is used in the problem of the stability of the translational motion of a rigid body in a uniform gravitational field when it undergoes collisions with a fixed horizontal plane and in the problem of the stability of one type of resonant in-plane rotations of a satellite, i.e., a rigid body, in an elliptic orbit.     

A splitting method for stochastic programs

This paper derives a new splitting-based decomposition algorithm for convex stochastic programs. It combines certain attractive features of the progressive hedging algorithm of Rockafellar and Wets, the dynamic splitting algorithm of Salinger and Rockafellar and an algorithm of Korf. We give two derivations of our algorithm. The first one is very simple, and the second one yields a preconditioner that resulted in a considerable speed-up in our numerical tests. The first author was supported by the Finnish Foundation for Economic Education under grants 20728 and 21599, and by Jenny ja Antti Wihuri Foundation.