We propose a linesearch projection algorithm for solving non-monotone and non-Lipschitzian equilibrium problems in Hilbert spaces. It is proved that the sequence generated by the proposed algorithm converges strongly to a solution of the equilibrium problem under the assumption that the solution set of the associated Minty equilibrium problem is nonempty. Compared with existing methods, we do not employ Fejér monotonicity in the strategy of proving the convergence. This comes from projecting a fixed point instead of the current point onto a subset of the feasible set at each iteration. Moreover, employing an Armijo-linesearch without subgradient has a great advantage in CPU-time. Some numerical experiments demonstrate the efficiency and strength of the presented algorithm.
This paper provides a comprehensive study of the nonmonotone forward-backward splitting (FBS) method for solving a class of nonsmooth composite problems in Hilbert spaces. The objective function is the sum of a Fréchet differentiable (not necessarily convex) function and a proper lower semicontinuous convex (not necessarily smooth) function. These problems appear, for example, frequently in the context of optimal control of nonlinear partial differential equations (PDEs) with nonsmooth sparsity promoting cost functionals. We discuss the convergence and complexity of FBS equipped with the nonmo…
This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problem in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo linesearch converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in ord…
In this paper, we propose new algorithms for finding a common point of the solution set of a pseudomonotone equilibrium problem and the set of fixed points of a symmetric generalized hybrid mapping in a real Hilbert space. The convergence of the iterates generated by each method is obtained under assumptions that the fixed point mapping is quasi-nonexpansive and demiclosed at $0$, and the bifunction associated with the equilibrium problem is weakly continuous. The bifunction is assumed to be satisfying a Lipschitz-type condition when the basic iteration comes from the extragradient method. It …
In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given …
Splitting Algorithms Using Pairs Monotonicity to Solve the Sum of Two Non-monotone Operators Problem
Ton Duc Thang University（胡志明市镏德胜大学）；Sidi Mohammed Ben Abdellah University, National School of Applied Sciences（西迪穆罕默德·本阿卜达拉大学国立应用科学学院）；University of Limoges（利摩日大学）
We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the struct…
The relationships between five classes of monotonicity, namely 3^*-, 3-cyclic, strictly, para-, and maximal monotonicity, are explored for linear operators and linear relations in Hilbert space. Where classes overlap, examples are given; otherwise their relationships are noted for linear operators in $\mathcal{R}^2$, $\mathcal{R}^n$, and general Hilbert spaces. Along the way, some results for linear relations are obtained.
Tseng's forward-backward-forward algorithm is a valuable alternative for Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich's method converges and can therefore be used also for solving variational inequalities governed by pseudo-monotone and Lipschitz continuous operators. In this paper, we first associate to a pseudo-monotone variational inequality a forward-backward-forward dynamical…
This paper is concerned with dynamic user equilibrium with elastic travel demand (E-DUE) when the trip demand matrix is determined endogenously. We present an infinite-dimensional variational inequality (VI) formulation that is equivalent to the conditions defining a continuous-time E-DUE problem. An existence result for this VI is established by applying a fixed-point existence theorem (Browder, 1968) in an extended Hilbert space. We present three algorithms based on the aforementioned VI and its re-expression as a differential variational inequality (DVI): a projection method, a self-adaptiv…
We consider monotone inclusions defined on a Hilbert space where the operator is given by the sum of a maximal monotone operator $T$ and a single-valued monotone, Lipschitz continuous, and expectation-valued operator $V$. We draw motivation from the seminal work by Attouch and Cabot on relaxed inertial methods for monotone inclusions and present a stochastic extension of the relaxed inertial forward-backward-forward (RISFBF) method. Facilitated by an online variance reduction strategy via a mini-batch approach, we show that (RISFBF) produces a sequence that weakly converges to the solution set…
In this paper we study the bounded perturbation resilience of projection and contraction algorithms for solving variational inequality (VI) problems in real Hilbert spaces. Under typical and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed projection and contraction algorithms is proved. Based on the bounded perturbed resilience of projection and contraction algorithms, we present some inertial projection and contraction algorithms. In addition we show that the perturbed algorithms converges at the rate of $O(1/t)$.
This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis relies on a notion of stochastic quasi-Fejér monotonicity, which is thoroughly investigated. The iterative methods under consideration feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. Algorithms using quasinonexpansive operators or compositions of averaged nonexpansive operators are cons…
In infinite-dimensional Hilbert spaces we device a class of strongly convergent primal-dual schemes for solving variational inequalities defined by a Lipschitz continuous and pseudomonote map. Our novel numerical scheme is based on Tseng's forward-backward-forward scheme, which is known to display weak convergence, unless very strong global monotonicity assumptions are made on the involved operators. We provide a simple augmentation of this algorithm which is computationally cheap and still guarantees strong convergence to a minimal norm solution of the underlying problem. We provide an adapti…
Non-Lipschitz Inertial Contraction-Type Method for Monotone Variational Inclusion problems
VIT Bhopal University（维特博帕尔大学）；Aligarh Muslim University（阿利加尔穆斯林大学）；China Medical University Hospital, China Medical University（中国医药大学附设医院，中国医药大学）；Academy of Romanian Scientists（罗马尼亚科学家学院）；Tiangong University（天津工业大学）
We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed poin…
The stochastic Auxiliary Problem Principle (APP) algorithm is a general Stochastic Approximation (SA) scheme that turns the resolution of an original optimization problem into the iterative resolution of a sequence of auxiliary problems. This framework has been introduced to design decomposition-coordination schemes but also encompasses many well-known SA algorithms such as stochastic gradient descent or stochastic mirror descent. We study the stochastic APP in the case where the iterates lie in a Banach space and we consider an additive error on the computation of the subgradient of the objec…
This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only one value of the mapping per iteration. This distinguishes our method from most other projection-type methods for variational inequalities with monotone mapping. Also we prove that it has R-linear rate of convergence und…
