## An analytic center cutting plane algorithm for finding equilibrium points

### Fernanda M. P. Raupp and Wilfredo Sosa

We apply an analytic center cutting plane algorithm in order to solve approximately the following Scalar Equilibrium Problem (SEP in short): find $\bar x \in K$ such that $f(\bar x,y) \geq 0$ for all $y \in K$, where K is a nonempty closed convex subset of $[0,1]^n$ and $f : [0,1]^n\times [0,1]^n \rightarrow \R$ is a function that is upper semicontinuous in the first variable, lower semicontinuous and convex in the second one. $f$ also satisfies the following properties: $f(x,x) = 0$ for all $x\in K$ and $f(x,y) \geq 0$ implies $f(y,x) \leq 0$. This problem is associated to a convex feasibility problem and has as particular cases: variational inequality problem, fixed point problem, complementarity problem, convex-concave constrained saddle point problem, convex minimization problem and Nash equilibria problem for noncooperative games, among others. We analyze the convergence and complexity of the algorithm, which is a variant of one in Goffin, Luo and Ye of 1996.

Technical Report 07/2000 LNCC - MCT - Brazil

Contact: fernanda@lncc.br