Qiao-Li Dong

Solving the split equality problem without prior knowledge of operator norms

The split equality problem has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Moudafi proposed an alternating CQ algorithm and its relaxed variant to solve it. However, to employ Moudafi’s algorithms, one needs to know a priori norm (or at least an estimate of the norm) of the bounded linear operators (matrices in the finite-dimensional framework). To estimate the norm of an operator is very difficult, but not an impossible task. It is the purpose of this paper to introduce a projection algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any priori information about the operator norms. We also practise this way of selecting stepsizes for variants of the projection algorithm, including a relaxed projection algorithm where the two closed convex sets are both level sets of convex functions, and a viscosity algorithm. Both weak and strong convergence are investigated.

Rate of convergence of Mann, Ishikawa and Noor iterations for continuous functions on an arbitrary interval

In this paper, we relax the control condition of convergence of SP-iteration presented by Phuengrattana and Suantai (J. Comput. Appl. Math. 235:3006-3014, 2011). We compare the rate of convergence of Mann, Ishikawa and Noor iterations from another point of view and come to a different conclusion. Finally, we provide a numerical example for Ishikawa and Noor iterations, which supports our theoretical results.MSC:47H05, 47H07, 47H10.

Approximately solving multi-valued variational inequalities by using a projection and contraction algorithm

A projection and contraction algorithm for solving multi-valued variational inequalities is proposed. The algorithm is proved to converge globally to a solution of a given multi-valued variational inequality under standard conditions. We present an analysis of the convergence rate. Finally, preliminary computational experiments illustrate the advantage of the proposed algorithm.

Simultaneous and semi-alternating projection algorithms for solving split equality problems

In this article, we first introduce two simultaneous projection algorithms for solving the split equality problem by using a new choice of the stepsize, and then propose two semi-alternating projection algorithms. The weak convergence of the proposed algorithms is analyzed under standard conditions. As applications, we extend the results to solve the split feasibility problem. Finally, a numerical example is presented to illustrate the efficiency and advantage of the proposed algorithms.

Bounded perturbation resilience of extragradient-type methods and their applications

In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI’s associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of O(1/t)

Totally relaxed, self-adaptive algorithm for solving variational inequalities over the intersection of sub-level sets

O(1/t)

Single projection method for pseudo-monotone variational inequality in Hilbert spaces

. Numerical illustrations are given to demonstrate the performances of the algorithms.

A modified subgradient extragradient method for solving the variational inequality problem

Abstract In this paper we study the classical Variational Inequality (VI) over the intersection of sub-level sets of finite convex functions in real Hilbert spaces. The Subgradient Extragradient method of Censor et al. extend Korpelevich’s extragradient method by introducing an additional constructible set and then there is the need to calculate only one orthogonal projection onto the feasible set per each iteration instead of two as in. Motivated by this result, we propose a new extension, called the Totally Relaxed and Self-adaptive Subgradient Extragradient Method, which does not require the calculation of any exact projections onto the VI’s feasible set. Hence, any general convex feasible sets can be involved in the VI, such as the finite intersection of sub-level sets of convex functions. In our new scheme we also introduce an adaptive step-size rule which avoids the need to know a priori the Lipschitz constant of the VI associated mapping. Under mild and standard assumptions, we prove weak convergence of the proposed method at rate in the ergodic sense. Two numerical examples are presented to illustrate the behaviour and performances of out proposed scheme.

Convergence of projection and contraction algorithms with outer perturbations and their applications to sparse signals recovery

ABSTRACT In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given.

A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. (J. Optim. Theory Appl. 148, 318–335, 2011), replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant.