Strong convergence of projected reflected gradient methods for variational inequalities. (English) Zbl 1460.65083
Summary: The purpose of this work is to revisit the numerical approach to classical variational inequality problems, with monotone and Lipschitz continuous mapping, by means of a regularized dynamical method. A main feature of the method is that it formally requires only one projection step onto the feasible set and only one evaluation of the involved mapping (at each iteration), combined with some viscosity-like regularization process. A strong convergence theorem is established in a general setting that allows the use of varying step-sizes without any requirement of additional projections. We also point out that the considered method in absence of regularization does not generate a Fejér-monotone monotone sequence. So a new analysis is developed for this purpose.
MSC:
| 65K15 | Numerical methods for variational inequalities and related problems |
| 49J40 | Variational inequalities |
| 47H05 | Monotone operators and generalizations |
| 90C25 | Convex programming |
| 90C30 | Nonlinear programming |
| 90C52 | Methods of reduced gradient type |
Keywords:
variational inequality; monotone operator; dynamical-type method; strong convergence; regularisation process; viscosity methodReferences:
| [1] | [1] D.P. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method, IEEE Trans. Automatic Control, 21(1976), no. 2, 174-184. · Zbl 0326.49025 |
| [2] | [2] D.P. Bertsekas, E.M. Gafni, Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Prog. Study, 17(1982), 139-159. 660PAUL-EMILE MAING ´E · Zbl 0478.90071 |
| [3] | [12] E.N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comp. Math. Phys., 27(1987), 120-127. · Zbl 0665.90078 |
| [4] | [13] G.M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 12(1976), 747-756. · Zbl 0342.90044 |
| [5] | [14] P.E. Maing´e, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis, 16(2008), no. 7-8, 899-912. · Zbl 1156.90426 |
| [6] | [15] P.E. Maing´e, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47(2008), no. 3, 1499-1515. · Zbl 1178.90273 |
| [7] | [16] P.E. Maing´e, M.L. Gobinddass, Convergence of one-step projected gradient methods for variational inequalities, Internal Report, Universit´e des Antilles, 2015. · Zbl 06661393 |
| [8] | [17] P.E. Maing´e, S. Maruster, Convergence in norm of Kranoselski-Mann iterations for fixed points of demicontractive mappings, Applied Mathematics and Computation, 217(2011), no. 24, 98649874. · Zbl 1226.65050 |
| [9] | [18] Yu. Malitski, Projected reflected gradient method for variational inequalities, SIAM J. Optim., 25(2015), no. 1, 502-520. · Zbl 1314.47099 |
| [10] | [19] Yu. Malitsky, V.V Semenov, An extragradient algorithm for monotone variational inequalities, Cybernetics and Systems Anal., 50(2014), 271-277. · Zbl 1311.49024 |
| [11] | [20] P. Marcotte, Applications of Khobotov’s algorithm to variational and network equlibrium problems, Inform. Systems Oper. Res., 29(1991), 258-270. · Zbl 0781.90086 |
| [12] | [21] M.A. Noor, Modified projection method for pseudomonotone variational inequlities, Applied Mathematics Letters, 15(2002), 315-320. · Zbl 1027.49005 |
| [13] | [22] M.V. Solodov, P. Tseng, Modified projection methods for monotone variational inequalities, SIAM J. Control Optim., 34(1996), no. 5, 1814-1834. · Zbl 0866.49018 |
| [14] | [23] M.V. Solodov, B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37(1999), 765-776. · Zbl 0959.49007 |
| [15] | [24] G. Stampacchia, Formes bilin´eaires coercitives sur les ensembles convexes, C.R.A.S., Paris, 258(1964), 4413-4416. · Zbl 0124.06401 |
| [16] | [25] D. Sun, An iterative method for solving variational inquality problems and complementarity problems, Numer. Math., J. Chin. Univ., 16(1994), 145-153. · Zbl 0817.65048 |
| [17] | [26] D. Sun, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory and Appl., 91(1996), 123-140. · Zbl 0871.90091 |
| [18] | [27] W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 1089.49500 |
| [19] | [28] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002 |
| [20] | [29] P. Tseng, On linear convergence of iterative methods for the variational inequality problem, J. Comp. Appl. Math., 60(1995), 237-252. · Zbl 0835.65087 |
| [21] | [30] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38(2000), 431-446. STRONG CONVERGENCE OF PROJECTED REFLECTED GRADIENT METHODS661 · Zbl 0997.90062 |
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