Bulletin of the
Korean Mathematical Society BKMS

A globally convergent algorithm for solving equilibrium problems is proposed. The algorithm is based on a proximal point algorithm (shortly $(PPA)$) with a positive definite matrix $M$ which is not necessarily symmetric. The proximal function in existing $(PPA)$ usually is the gradient of a quadratic function, namely, $\bigtriangledown (\|x\|^2_M)$. This leads to a proximal point-type algorithm. We first solve pseudomonotone equilibrium problems without Lipschitzian assumption and prove the convergence of algorithms. Next, we couple this technique with the Banach contraction method for multivalued variational inequalities. Finally some computational results are given.

Keywords: equilibrium problems, proximal point algorithm, pseudomonotonicity, linear proximal function, Banach contraction method

MSC numbers: 65K10, 90C25