MULTIPLICATIVE INTERIOR GRADIENT METHODS FOR MINIMIZATION OVER THE NONNEGATIVE ORTHANT

We introduce a new class of multiplicative iterative methods for solvi ng minimization problems over the nonnegative orthant. The algorithm i s akin to a natural extension of gradient methods for unconstrained mi nimization problems to the case of nonnegativity constraints, with the special feature that it generates a sequence of iterates which remain in the interior of the nonnegative orthant. We prove that the algorit hm combined with an appropriate line search is weakly convergent to a saddle point of the minimization problem, when the minimand is a diffe rentiable function with bounded level sets. If the function is convex, then weak convergence to an optimal solution is obtained. Moreover, b y using an appropriate regularized line search, we prove that the leve l set boundeness hypothesis can be removed, and full convergence of th e iterates to an optimal solution is established in the convex case.