On Bilevel Variational Inequalities
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On Bilevel Variational Inequalities Zhongping Wan · Jia-wei Chen
Received: 13 May 2013 / Revised: 15 November 2013 / Accepted: 7 December 2013 / Published online: 18 December 2013 © Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg 2013
Abstract A class of bilevel variational inequalities (shortly (BVI)) with hierarchical nesting structure is firstly introduced and investigated. The relationship between (BVI) and some existing bilevel problems are presented. Subsequently, the existence of solution and the behavior of solution sets to (BVI) and the lower level variational inequality are discussed without coercivity. By using the penalty method, we transform (BVI) into one-level variational inequality, and establish the equivalence between (BVI) and the one-level variational inequality. A new iterative algorithm to compute the approximate solutions of (BVI) is also suggested and analyzed. The convergence of the iterative sequence generated by the proposed algorithm is derived under some mild conditions. Finally, some relationships among (BVI), system of variational inequalities and vector variational inequalities are also given. Keywords Bilevel variational inequalities · Bilevel programs · System of variational inequality · Vector variational inequality · Penalty method Mathematics Subject Classification (2000) 90C33 · 49J40 · 47H05
This work was supported by the Natural Science Foundation of China (Nos. 71171150, 11201039), the Doctor Fund of Southwest University (No. SWU113037) and the Fundamental Research Funds for the Central Universities (No. XDJK2014C073). Z. Wan School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P.R. China e-mail: [email protected]
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J.-w. Chen ( ) School of Mathematics and Statistics, Southwest University, Chongqing 400715, P.R. China e-mail: [email protected]
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Z. Wan, J.-w. Chen
1 Introduction Variational inequalities, which were introduced by Stampacchia [51] in 1964, have been intensively studied and widely applied to some practical problems arising in economics, transportation, network and structural analysis, elasticity, engineering and mechanics, supply chain management, finance and game theory (see, e.g., [10–12, 28, 33–36, 38, 39, 47, 49, 54, 55, 58, 61] etc.). Recently, optimization problem with variational inequality, equilibrium and complementarity constraints, have caused many authors’ interests (see, e.g., [7, 18, 31, 32, 40–43, 45, 50, 56, 59, 60] etc.). Mordukhovich [44] studied an equilibrium problems with equilibrium constraints (EPECs) and mathematical programs with equilibrium constraints (MPECs) via multiobjective optimization, and obtained some optimality conditions for (EPECs) based on advanced generalized differential tools of variational analysis. Outrata [48] applied the (EPECs) to an oligopolistic market with several large and several small firms, derived two types of necessary conditions for a solution of this game and briefly discussed the possibilities of it
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