Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems

• Main Authors: Liou Yeong-Cheng, Chen Chia-Ping, Yao Yonghong
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2010/642584

<p/> <p>We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point <inline-formula> <graphic file="1687-1812-2010-642584-i1.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1812-2010-642584-i2.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2010-642584-i3.gif"/></inline-formula>, where <inline-formula> <graphic file="1687-1812-2010-642584-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-642584-i5.gif"/></inline-formula> are two monotone operators and <inline-formula> <graphic file="1687-1812-2010-642584-i6.gif"/></inline-formula> is the solution of the equilibrium problem of finding <inline-formula> <graphic file="1687-1812-2010-642584-i7.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1812-2010-642584-i8.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2010-642584-i9.gif"/></inline-formula>. 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 <inline-formula> <graphic file="1687-1812-2010-642584-i10.gif"/></inline-formula>. We prove that the net <inline-formula> <graphic file="1687-1812-2010-642584-i11.gif"/></inline-formula> hierarchically converges to the solution of (HEVP); that is, for each fixed <inline-formula> <graphic file="1687-1812-2010-642584-i12.gif"/></inline-formula>, the net <inline-formula> <graphic file="1687-1812-2010-642584-i13.gif"/></inline-formula> converges in norm, as <inline-formula> <graphic file="1687-1812-2010-642584-i14.gif"/></inline-formula>, to a solution <inline-formula> <graphic file="1687-1812-2010-642584-i15.gif"/></inline-formula> of the equilibrium problem, and as <inline-formula> <graphic file="1687-1812-2010-642584-i16.gif"/></inline-formula>, the net <inline-formula> <graphic file="1687-1812-2010-642584-i17.gif"/></inline-formula> converges in norm to the unique solution <inline-formula> <graphic file="1687-1812-2010-642584-i18.gif"/></inline-formula> of (HEVP).</p>