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This paper aims to develop a Newton-type method to solve a class of nonconvexcomposite programs. In particular, the nonsmooth part is possibly nonconvex. Totackle the nonconvexity, we develop a notion of strong prox-regularity which isrelated to the singleton property, Lipschitz continuity, and monotonicity ofthe associated proximal operator, and we verify it in various classes offunctions, including weakly convex functions, indicator functions of proximallysmooth sets, and two specific sphere-related nonconvex nonsmooth functions. Inthis case, the problem class we are concerned with covers smooth optimizationproblems on manifold and certain composite optimization problems on manifold.For the latter, the proposed algorithm is the first second-order type method.Combining with the semismoothness of the proximal operator, we design aprojected semismooth Newton method to find a root of the natural residualinduced by the proximal gradient method. Since the corresponding naturalresidual may not be globally monotone, an extra projection is added on theusual semismooth Newton step and new criteria are proposed for the switchingbetween the projected semismooth Newton step and the proximal step. The globalconvergence is then established under the strong prox-regularity. Based on theBD regularity condition, we establish local superlinear convergence. Numericalexperiments demonstrate the effectiveness of our proposed method compared withstate-of-the-art ones.

A projected semismooth Newton method for a class of nonconvex composite programs with strong prox-regularity