A first-order regularized algorithm with complexity properties for the unconstrained and the convexly constrained low order-value optimization problem

In this paper we consider the minimization of the unconstrained low order-value function. We also consider the case in which the feasible region is given by a closed convex set, assuming that the projection operation is affordable. For both cases, we introduce regularized first-order algorithms and prove worst-case iteration and evaluation complexity results. Asymptotic convergence results are also presented. The proposed algorithm for the case of constraints given by an arbitrary closed convex set has the classical projected gradient method as a particular case. The algorithms are implemented and several numerical examples illustrate their application. From a theoretical point of view, there is no method for the low order-value problem that has a complexity analysis and for which the relation between complexity results and asymptotic results has been analyzed. From a practical point of view, one of the applications considered is the training of a neural network. In this example, it is shown that the introduced method outperforms another recently introduced method that represents the state of the art for solving low order-value problems.