Before revision:  Starting in a neutral strategy with constant exercise rate then allows us to globally optimize this function in a gradual manner . Numerical experiments on vanilla put options in the multivariate Black--Scholes model and preliminary theoretical analysis underline the efficiency of our method  both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of  exercise rates.
After revision:  The global optimum of this function can be approached gradually when starting from a constant exercise rate then allows us to globally optimize this function in a gradual manner . Numerical experiments on vanilla put options in the multivariate Black--Scholes model and preliminary theoretical analysis underline the efficiency of our method both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of exercise rates.