Solving a class of linear projection equations

Summary.

Many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation$u = P_{\Omega} [ u - (Mu+q)] .$ Here, $P_{\Omega}(\cdot)$ is the orthogonal projection on some convex set$\Omega$ (e.g. $\Omega= {\Bbb R}^n_+$) and $M$ is a positive semidefinite matrix. This paper presents some new methods for solving a class of linear projection equations. The search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization. Based on the idea of a projection and contraction method [7] we get a simple step length rule and are able to obtain global linear convergence.