On the properties of solutions to a differential inclusion associated with a nonsmooth constrained optimization problem

In this paper we consider, under very general conditions, a constrained minimization problem and we associate to this problem a differential inclusion which has the property that all the trajectories converge to the set C of constrained critical points. The conditions on the functional to be minimized and on the function which defines the constraint are the minimal requirements on these data to use the tools of the nonsmooth analysis to show the convergence of the trajectories to C. Furthermore, if these functions are also subanalytic, then it is proved that any trajectory converges to a critical point and it has finite length. In fact, we show that these assumptions guarantee that the multivalued vector field defining the differential inclusion satisfies a Lojasiewicz-type inequality. The dependence of the rate of convergence on the values of the Lojasiewicz exponent is also shown.