Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems
Yonghong Yao, Yeong‐Cheng Liou, Chia-Ping Chen · Fixed Point Theory and Applications · 2010
Abstract We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point "Equation missing" such that "Equation missing", for all "Equation missing", where "Equation missing", "Equation missing" are two monotone operators and "Equation missing" is the solution of the equilibrium problem of finding "Equation missing" such that "Equation missing", for all "Equation missing". We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net "Equation missing". We prove that the net "Equation missing" hierarchically converges to the solution of (HEVP); that is, for each fixed "Equation missing", the net "Equation missing" converges in norm, as "Equation missing", to a solution "Equation missing" of the equilibrium problem, and as "Equation missing", the net "Equation missing" converges in norm to the unique solution "Equation missing" of (HEVP).