Exponential stability of partial primal–dual gradient dynamics with nonsmooth objective functions

In this paper, we investigate the continuous time partial primal–dual gradient dynamics (P-PDGD) for solving convex optimization problems with form minx?X,y??f(x)+h(y),s.t.Ax+By=C, where f(x) is strongly and smooth, but h(y) non-smooth. Affine equality general set constraints are included. We prove existence of solution to P-PDGD its exponential stability. Then, bounds on decaying rates provided. Moreover, it also shown that can be regulated by setting stepsize.