Abstract
The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.
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References
Auslender A, Haddou M (1995) An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Program 71(1):77–100
Auslender A, Teboulle M (2000) Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J Optim 10(4):1097–1115
Auslender A, Teboulle M, Ben-Tiba S (1999) A logarithmic-quadratic proximal method for variational inequalities. Comput Optim Appl 12:31–40
Bertsekas DP, Tsitsiklis JN (1989) Parallel and distributed computation: numerical methods. Prentice-Hall, Englewood Cliffs
Burachik RS, Iusem AN (1998) A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J Optim 8:197–216
Censor Y, Iusem AN, Zenios SA (1998) An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math Program 81:373–400
Eckstein J (1998) Approximate iterations in Bregman-function-based proximal algorithms. Math Program 83:113–123
Güler O (1991) On the convergence of the proximal point algorithm for convex minimization. SIAM J Con Optim 29(2):403–419
He BS, Liao L-Z (2002) Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl 112(1):111–128
He BS, Liao L-Z, Yuan X-M (2006) A LQP based interior prediction-correction method for nonlinear complementarity problems. J Comput Math 24(1):33–44
Khobotov EN (1987) Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput Math Phys 27:120–127
Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Economics Math Methods 12:747–756
Martinet B (1970) Regularisation d’ inequations variationelles par approximations sucessives. Rev Francaise Inf Rech Oper 4:154–159
Nagurney A, Zhang D (1996) Projected dynamical systems and variational inequalities with applications. Kluwer Boston
Nocedal J, Wright SJ (1999) Numerical optimization. Springer, Heidelberg
Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Con Optim 14:877–898
Teboulle M (1997) Convergence of proximal-like algorithms. SIAM J Optim 7(4):1069–1083
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Li, M., Shao, H. & He, B.S. An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities. Math Meth Oper Res 66, 183–201 (2007). https://doi.org/10.1007/s00186-007-0145-1
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DOI: https://doi.org/10.1007/s00186-007-0145-1