Collection of selected papers from International Workshop on Nonlinear and Variational Analysis, Tianjin, China 2019
This special issue of the journal “Optimization” is dedicated to the ‘International Workshop on Nonlinear and Variational Analysis 2019’, which was held during July 15–17, 2019, at Tianjin Polytechnic University, Tianjin, China. During the workshop, inspiring keynote speeches along with various interesting research presentations by highly esteemed researchers allowed for a stimulating environment to discuss the latest research on nonlinear and variational analysis. The workshop presented an ideal surrounding for new collaboration opportunities and exchanging ideas, which, among others, led to the contributions that are provided in this special issue.
The objective of this special issue is to present advances in the growing research areas in optimization from the theoretical as well as from the application point of view. In particular, this special issue addresses different areas of applied analysis and optimization theory, recent results in nonlinear analysis, variational analysis, optimal control, fixed point theory, and corresponding applications. It is our great pleasure to summarize the novel contributions of the finally chosen manuscripts below.
The paper ‘Iterative methods for fixed points and zero points of nonlinear mappings with applications’ by L. Liu, X. Qin and R.P. Agarwal investigates a descent-type iterative algorithm for finding a common element of fixed-point sets of a finite family of nonexpansive mappings and zero-point sets of pseudomonotone mappings in Hilbert spaces. With suitable assumptions, necessary and sufficient conditions for the strong convergence of the algorithm are derived. Numerical examples and applications to signal processing problems are provided to support the main results.
In an exciting contribution entitled ‘Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings’, the authors L.-C. Ceng and M. Shang introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, the authors prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.
The paper ‘A method with inertial extrapolation step for split monotone inclusion problems’ by Y. Yao, Y. Shehu, X.-H. Li and Q.-L. Dong presents convergence analysis of an iterative algorithm with inertial extrapolation step for finding an approximate solution of split monotone inclusion problem in real Hilbert spaces. Weak convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. Some special cases of the general problem are given and the authors present some numerical implementations to support the theoretical analysis and give justification for the addition of the extrapolation step in the proposed method.
The paper ‘Graph contractions in vector-valued metric spaces and applications’ by A. Petruşel, G. Petruşel and J.-C. Yao present a study of a fixed point equation for a large class of operators by the following perspectives: existence, uniqueness, approximation, data dependence of the operator perturbation, well-posedness, and Ulam-Hyers stability. The non-self case is also discussed and some applications are given.
In another valuable contribution, entitled ‘New regularity conditions and Fenchel dualities for DC optimization problems involving composite functions’, the authors D. Fang, Q.H. Ansari and J.-C. Yao consider a DC composite optimization problem. By using the properties of the epigraph of the conjugate functions, they introduce some new regularity conditions and obtain complete characterizations for weak / strong/stable Fenchel dualities and for zero / stable zero duality gap properties of the considered problem.
The manuscript ‘A new self-adaptive method for the split equality common fixed-point problem of quasi-nonexpansive mappings’, authored by J. Zhao, D. Hou and X. Wang, introduces a new iterative algorithm from primal-dual methods for solving the split equality common fixed-point problem of quasi-nonexpansive mappings in real Hilbert space. The proposed algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. The authors use a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about bounded linear operator norms. It avoids the difficult task of estimating the operator norms. Under suitable conditions, they obtain the weak convergence of the proposed algorithm. The performance of the proposed algorithm is also illustrated by preliminary numerical experiments.
In the paper ‘On projected alternating BB methods for variational inequalities’ by Y. Qu, W. Tong, H. He & Y.-C. Liou, the authors follow the spirit of Barzilai–Borwein (BB) step size and propose a projection method with alternate BB step size for general variational inequalities. Although the global convergence is established under some strong conditions, a series of computational experiments on nonlinear complementarity problems, image deblurring problems and generalized Nash equilibrium problems demonstrate that the proposed almost-parameter-free projection method is more efficient than some existing state-of-the-art projection methods in the literature.
In the paper ‘Characterizations of robust ϵ-quasi optimal solutions for nonsmooth optimization problems with uncertain data’ by X.-K. Sun, K. L. Teo and X.-J. Long, the authors deals with robust ϵ-quasi optimal solutions for a class of nonsmooth optimization problems with uncertain data. Under some mild assumptions, they first establish, by using robust optimization (i.e. worst-case) approach, approximate optimality conditions for this uncertain nonsmooth optimization problem. Then, Mixed-type robust approximate dual problem of this uncertain optimization problem is introduced, and their relationships are explored. Moreover, using a scalarization method, optimality conditions for robust weakly approximate efficient solutions for an uncertain nonsmooth multiobjective optimization problem are derived. Moreover, the authors also obtain approximate duality theorems for the uncertain nonsmooth multiobjective optimization problem.
Finally, the paper ‘Two projection algorithms for a class of split feasibility problems with jointly constrained Nash equilibrium models’ by Q.L. Dong, X.-H. Li and T.M. Rassias concludes this special issue. In this work, the authors consider a certain split feasibility problem. Two projection algorithms for solving the SFP by combining the projection method for the equilibrium problem are introduced and the gradient method for a considered inclusion is derived. The proposed algorithms converge to a solution of the SFP under weak conditions. Moreover, the authors present a numerical example for a jointly constrained Nash equilibrium model in electricity production market to demonstrate the behaviour of the proposed algorithms.
The guest editors are grateful to everyone who contributed and supported to this special issue. In particular, we would like to thank all the authors who contributed to this special issue and all the reviewers who kindly accepted the invitation to provide their expertise and gave constructive comments. We are especially grateful to the Editor in Chief of ‘Optimization: A Journal of Mathematical Programming and Operations Research’, Prof. Dr. Christiane Tammer, for giving us the editorial opportunity to organize this special issue, and we are deeply indebted for her expertise, advice, and constant support.
Guest Editors
Elisabeth Köbis ([email protected], Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Norway)
Jinlu Li ([email protected], Department of Mathematical Sciences, Shawnee State University, USA)
Adrian Petruşel ([email protected], Faculty of Mathematics and Computer Science, Babeş-Bolyai University Cluj-Napoca, Romania)
Jen-Chih Yao ([email protected], Center for General Education, China Medical University, Taichung, Taiwan; Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan)