A generalized Persidskii theorem and its applications to nonsmooth gradient dynamical systems

A generalized Persidskii-like theorem is derived and shown to be applicable to the stability analysis of a class of gradient dynamical systems with discontinuous right hand sides. These dynamical systems arise from the steepest descent technique applied to a variety of problems suitably formulated as constrained minimization problems. The problems susceptible to this approach include linear programming problems and specifically the k-winners-take-all problem, the problem of solving underdetermined linear systems arising in least squares support vector machines, and quadratic programming problems associated to the support vector machine approach to classification. The advantage of the proposed analysis is the derivation of simply computable convergence conditions.