An algorithm for the general nonliner constrained optimization problem is developed. This algorithm consists of a sequence of conjugate gradient- restoration cycles. The conjugate gradient portion of each cycle is based upon a conjugate gradient algorithm that is derived for the special case of a quadratic function subject to linear constraints. This portion of the cycle involves a single step and is designed to decrease the value of the function while satisfying the constraints to first order. The restoration portion of each cycle involves one or more iterations and is designed to restore the norm of the constraint function to within a predetermined tolerance about zero. In order to accelerate the convergence of the algorithm a positive-definite preconditioning matrix is used. Three numerical examples are also given.