A Cutting Hyperplane Projection Method for Solving Generalized Quasi-Variational Inequalities
- PDF / 531,424 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 75 Downloads / 175 Views
A Cutting Hyperplane Projection Method for Solving Generalized Quasi-Variational Inequalities Ming-Lu Ye1
Received: 24 August 2015 / Revised: 23 November 2015 / Accepted: 24 February 2016 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2016
Abstract The generalized quasi-variational inequality is a generalization of the generalized variational inequality and the quasi-variational inequality. The study for the generalized quasi-variational inequality is mainly concerned with the solution existence theory. In this paper, we present a cutting hyperplane projection method for solving generalized quasi-variational inequalities. Our method is new even if it reduces to solve the generalized variational inequalities. The global convergence is proved under certain assumptions. Numerical experiments have shown that our method has less total number of iterative steps than the most recent projection-like methods of Zhang et al. (Comput Optim Appl 45:89–109, 2010) for solving quasi-variational inequality problems and outperforms the method of Li and He (J Comput Appl Math 228:212–218, 2009) for solving generalized variational inequality problems. Keywords Generalized quasi-variational inequality · Cutting hyperplane projection method · Point-to-set mapping · Pseudomonotone Mathematics Subject Classification
47J20 · 49J40
This research was supported by the Scientific Research Foundation of Education Department of Sichuan Province (No. 15ZA0154), Scientific Research Foundation of China West Normal University (No. 14E014), University Innovation Team Foundation of China West Normal University (No. CXTD2014-4), and the National Natural Science Foundation of China (No. 11371015).
B 1
Ming-Lu Ye [email protected] College of Mathematics and Information, China West Normal University, Nanchong 637002, Sichuan, China
123
M.-L. Ye
1 Introduction Let C ⊂ Rn be a nonempty closed convex set. The generalized quasi-variational inequality (GQVI(F,K ,C)) problem is to find a point x ∗ ∈ K (x ∗ ) and a point ξ ∈ F(x ∗ ) such that (1.1) ξ, y − x ∗ 0, ∀ y ∈ K (x ∗ ), where F : C → 2R is a point-to-set mapping with nonempty compact convex values, K : C → 2C is a point-to-set mapping with nonempty closed and convex values, and ·, · denotes the usual inner product in Rn . Let Sˆ := {x ∈ K (x)|∃ ξ ∈ F(x) such that ξ, y − x 0 ∀ y ∈ K (x)}. If for all x ∈ C, K (x) ≡ C, i.e., K (x) is a constant set, then the GQVI(F,K ,C) problem reduces to the generalized variational inequality(GVI(F,C)) problem, i.e., to find a point x ∗ ∈ C and a point ξ ∈ F(x ∗ ) such that n
ξ, y − x ∗ 0, ∀ y ∈ C.
(1.2)
If F is a point-to-point mapping, then the GQVI(F,K ,C) problem reduces to the quasi-variational inequality(QVI(F,K ,C)) problem, i.e., to find a point x ∗ ∈ K (x ∗ ) such that (1.3) F(x ∗ ), y − x ∗ 0, ∀ y ∈ K (x ∗ ). If K (x) ≡ C and F is a point-to-point mapping, then the GQVI(F,K ,C) problem reduces to the variational inequality (VI(F,C)) pr
Data Loading...
