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

Abstract

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