Inertial extragradient algorithms for strongly pseudomonotone variational inequalities

Abstract The purpose of this paper is to study and analyze two different kinds of inertial type iterative methods for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The projection method is used to design the algorithms which can be computed more easily. The construction of solution approximations and the proof of convergence of the algorithms are performed without prior knowledge of the modulus of strong pseudomonotonicity and the Lipschitz constant of cost operator. Instead of that, the algorithms use variable stepsize sequences which are diminishing and non-summable. The numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with several previously known algorithms.

[1]  P. Maingé.  Inertial Iterative Process for Fixed Points of Certain Quasi-nonexpansive Mappings , 2007 .

[2]  Paul-Emile Maingé,et al.  Convergence of One-Step Projected Gradient Methods for Variational Inequalities , 2016, J. Optim. Theory Appl..

[3]  Duong Viet Thong,et al.  An inertial method for solving split common fixed point problems , 2017 .

[4]  Dang Van Hieu.  Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems , 2017, Numerical Algorithms.

[5]  Yu. V. Malitsky,et al.  A hybrid method without extrapolation step for solving variational inequality problems , 2015, J. Glob. Optim..

[6]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[7]  Yair Censor,et al.  Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space , 2012 .

[8]  P. Maingé.  Regularized and Inertial algorithms for common fixed points of nonlinear operators , 2008 .

[9]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[10]  Dang Van Hieu,et al.  Parallel Extragradient-Proximal Methods for Split Equilibrium Problems , 2015, 1511.02474.

[11]  A. Moudafi,et al.  Convergence of a splitting inertial proximal method for monotone operators , 2003 .

[12]  Qingzhi Yang.  On variable-step relaxed projection algorithm for variational inequalities , 2005 .

[13]  Pham Ky Anh,et al.  Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings , 2015, Numerical Algorithms.

[14]  M. Mathematical,et al.  Weak Convergence Theorem by an Extragradient Method for Variational Inequality, Equilibrium and Fixed Point Problems , 2009 .

[15]  Pham Ky Anh,et al.  Modified hybrid projection methods for finding common solutions to variational inequality problems , 2017, Comput. Optim. Appl..

[16]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[17]  Jen-Chih Yao,et al.  Solution Methods for Pseudomonotone Variational Inequalities , 2008 .

[18]  Jen-Chih Yao,et al.  On the solution existence of pseudomonotone variational inequalities , 2008, J. Glob. Optim..

[19]  Phan Tu Vuong,et al.  Modified projection method for strongly pseudomonotone variational inequalities , 2014, J. Glob. Optim..

[20]  Duong Viet Thong,et al.  New extragradient-like algorithms for strongly pseudomonotone variational inequalities , 2018, J. Glob. Optim..

[21]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[22]  I. Konnov.  Combined Relaxation Methods for Variational Inequalities , 2000 .

[23]  G. Stampacchia,et al.  On some non-linear elliptic differential-functional equations , 1966 .

[24]  Duong Viet Thong.  Viscosity approximation methods for solving fixed-point problems and split common fixed-point problems , 2017 .

[25]  Yair Censor,et al.  Algorithms for the Split Variational Inequality Problem , 2010, Numerical Algorithms.

[26]  Thomas Brox,et al.  iPiasco: Inertial Proximal Algorithm for Strongly Convex Optimization , 2015, Journal of Mathematical Imaging and Vision.

[27]  Yu. V. Malitsky,et al.  Projected Reflected Gradient Methods for Monotone Variational Inequalities , 2015, SIAM J. Optim..

[28]  R. Boţ,et al.  An inertial alternating direction method of multipliers , 2014, 1404.4582.

[29]  Yair Censor,et al.  Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space , 2011, Optim. Methods Softw..

[30]  Phayap Katchang,et al.  A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings , 2009 .

[31]  Duong Viet Thong,et al.  Weak and strong convergence theorems for variational inequality problems , 2017, Numerical Algorithms.

[32]  Dirk A. Lorenz,et al.  An Inertial Forward-Backward Algorithm for Monotone Inclusions , 2014, Journal of Mathematical Imaging and Vision.

[33]  R. U. Verma.  Variational inequalities involving strongly pseudomonotone hemicontinuous mappings in nonreflexive Banach spaces , 1998 .

[34]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[35]  Duong Viet Thong,et al.  Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems , 2018 .

[36]  Dang Van Hieu,et al.  New extragradient method for a class of equilibrium problems in Hilbert spaces , 2018 .

[37]  Paul-Emile Maingé.  Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure , 2016, Comput. Math. Appl..

[38]  Radu Ioan Bot,et al.  An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems , 2014, Numerical Algorithms.

[39]  Juan Peypouquet,et al.  A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization , 2014, SIAM J. Optim..

[40]  H. Attouch,et al.  An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping , 2001 .

[41]  P. Kumam,et al.  Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization , 2009 .

[42]  R. Boţ,et al.  A Hybrid Proximal-Extragradient Algorithm with Inertial Effects , 2014, 1407.0214.

[43]  Y. J. Cho,et al.  Inertial projection and contraction algorithms for variational inequalities , 2018, J. Glob. Optim..

[44]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[45]  Do Sang Kim,et al.  Qualitative properties of strongly pseudomonotone variational inequalities , 2016, Optim. Lett..

[46]  I. Konnov.  Equilibrium Models and Variational Inequalities , 2013 .

[47]  E. U. Ofoedu,et al.  Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space , 2006 .

[48]  Shiqian Ma,et al.  Inertial Proximal ADMM for Linearly Constrained Separable Convex Optimization , 2015, SIAM J. Imaging Sci..

[49]  Duong Viet Thong,et al.  Modified subgradient extragradient method for variational inequality problems , 2017, Numerical Algorithms.

[50]  Pham Duy Khanh,et al.  A New Extragradient Method for Strongly Pseudomonotone Variational Inequalities , 2016 .

[51]  Yair Censor,et al.  The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space , 2011, J. Optim. Theory Appl..

[52]  Radu Ioan Bot,et al.  Inertial Douglas-Rachford splitting for monotone inclusion problems , 2014, Appl. Math. Comput..