We introduce a geometrically transparent strict saddle property for nonsmooth functions. This guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. argue the may be realistic assumption in applications, since it provably holds generic semi-algebraic optimization problems.

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 provide...

Matrix-valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized differential properties of such functions related to nonsmooth-smoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher’s theorem, -derivative, directional derivative, and sem...

Matrix valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized differential properties of such functions related to nonsmooth-smoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher’s theorem, B-derivative, directional derivative, and se...

In this paper, we propose a redistributed proximal bundle method for class of nonconvex nonsmooth optimization problems with inexact information, i.e., consider the problem computing approximate critical points when only information about function values and subgradients are available show that reasonable convergence properties obtained. We assume errors in computation functions bounded princip...