We present a new projection method for solving quasi-variational inequalities. This new method consists of following two steps: First, we construct a hyperplane which can separate strictly current iterate from the solution set of the quasi-variational inequalities. Second, the next iterate is generated by projecting the current iterate onto the intersection of the feasible set C and this hyperplane. Our method is proven to be globally convergent under certain assumptions. Nnumerical experiments show that our method have the less total number of iterative steps.
YE Minglu. A Projection Method for Solving Quasi-variational Inequalities[J]. Acta Mathematicae Applicatae Sinica, 2014, 37(1): 160-169.
Wang Y J, Xiu N H, Wang C Y. Unified Framework of Extragradient-type Methods for Pseudomonotone Variational Inequalities. Journal of Optimization Theory and Applications, 2001, 111(3): 641-656
Gesztesy F, Simon B. On the Determination of a Potential from Three Spectra. Transactions of the American Mathematical Society, 1999, 189: 85-92
Fukushima M. A Successive Quadratic Programming Algorithm with Global and Superlinear Convergence Properties. Mathematical Programming, 1986, 35: 253-264
He Y R. A New Double Projection Algorithm for Variational Inequalities. Journal of Computational and Applied Mathematics, 2006, 185(1): 166-173
叶明露. 变分不等式的一类二次投影算法. 应用数学学报, 2012, 35(3): 529-535(Ye M L. The Framework of Double Projection Algorithm for Variational Inequalities. Acta Mathematicae Applicatae Sinica, 2012, 35(3): 529-535)
Facchinei F, Lucidi S. Quadraticly and Superlinearly Convergent for the Solution of Inequality Constrained Optimization Problem. Journal of Optimization Theory and Applications, 1995, 85(2): 265-289
Pivovarchik V N. An Inverse Sturm-liouville Problems by Three Spectra. Integral Equations and Operator Theory, 1999, 34: 234-243
Xiu N H, Zhang J Z. Some Recent Advances in Projection-type Methods for Variational Inequalities. Journal of Computational and Applied Mathematics, 2003, 152(1-2): 559-585
Fu S Z, Xu Z B, Wei G S. The Interlacing of Spectra Between Contionuous and Discontinuous Sturm-liouville Problems and Its Application to Inverse Problems. Taiwanese Journal of Mathematics, 2012, 16(2): 651-663
Facchinei F, Pang J S. Finite-dimensional Variational Inequalities and Complementarity Problems Volume I. New York: Springer-Verlag, 2003
Gasymov M G, Levitan B M. Determining the Dirac System from Scattering Phase Dokl. Akad. Nauk SSSR, 1966, 167(6): 1219-1222
Outrata J, Zowe J. A Numerical Approach to Optimization Problems with Variational Inequality Constraints. Mathematical Programming, 1995, 68(1-3): 105-130