Some recent advances in projection-type methods for variational inequalities

Xiu, Naihua; Zhang, Jianzhong

doi:10.1016/s0377-0427(02)00730-6

Cited by 108 publications

(42 citation statements)

References 104 publications

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“…Secondly, it is able to drop and add many constraints from and to the active set at each iteration. Due to its structural and theoretical advantages, various projection-type methods [8,9,12,21,25,27,28], such as the basic projection algorithm, the extragradient algorithm and its variants, and the hyperplane projection algorithm, have been designed to solve different convex optimization problems or monotone variational inequality problems. Interested readers may consult the monograph by Facchinei and Pang [5].…”

Section: Introductionmentioning

confidence: 99%

Some projection-like methods for the generalized Nash equilibria

Zhang

Xiu

2008

Comput Optim Appl

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Section: Introductionmentioning

confidence: 99%

Some projection-like methods for the generalized Nash equilibria

Zhang

Xiu

2008

Comput Optim Appl

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“…In the setting of Hilbert spaces, one of the most efficient numerical techniques is the projection method and its variant forms; see 4,[6][7][8][9][10][11][12][13][14][15] . Since the standard projection method strictly depends on the inner product property of Hilbert spaces, it can no longer be applied for general mixed type variational inequalities in Banach spaces.…”

Section: Fixed Point Theory and Applicationsmentioning

confidence: 99%

Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space

Ceng

Kum

Yao

2010

Fixed Point Theory Appl

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Let B be a real Banach space with the dual space B * . Let φ : B → R ∪ { ∞} be a proper functional and let Θ : B × B → R be a bifunction. In this paper, a new concept of η-proximal mapping of φ with respect to Θ is introduced. The existence and Lipschitz continuity of the η-proximal mapping of φ with respect to Θ are proved. By using properties of the η-proximal mapping of φ with respect to Θ, a generalized mixed equilibrium problem with perturbation for short, GMEPP is introduced and studied in Banach space B. An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space B, and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in B H a Hilbert space.

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“…These algorithms were also proposed for solving variational inequality problems, see f.i. [22]. In this work, we established the expected exact equivalence of non-degenerate LP problems and specific projection problems.…”

Section: Introductionmentioning

confidence: 97%