On the convergence properties of scaled gradient projection methods with non-monotone Armijo–like line searches

In order to solve constrained optimization problems on convex sets, the class of scaled gradient projection methods is often exploited in combination with non-monotone Armijo–like line search strategies. These techniques are adopted for efficiently selecting the steplength parameter and can be realized by means of two different approaches: either the one along the arc or the one along the feasible directions. In this paper we deeply analyze the convergence properties of the scaled gradient projection methods equipped with the non-monotone version of both these Armijo–like line searches. To the best of our knowledge, not all the convergence results proved for either the non-scaled or the monotone gradient projection algorithm have been also stated for the non-monotone and scaled counterpart. The goal of this paper is to fill this gap of knowledge by detailing which hypotheses are needed to guarantee both the stationarity of the limit points and the convergence of the sequence generated by the non-monotone scaled gradient projection schemes. Moreover, in the case of polyhedral constraint set, we discuss the identification of the active set at the solution for the sequence generated by the along the arc approach. Several numerical experiments on quadratic and non-quadratic optimization problems have been carried out in order to compare the behaviour of the considered scaled gradient projection methods.