Variational Properties of Decomposable Functions. Part II: Strong Second-Order Theory

Abstract

Local superlinear convergence of the semismooth Newton method usually necessitates assumptions on the uniform invertibility of the utilized, generalized Jacobian matrices, such as, e.g., BD- or CD-regularity. For certain composite-type problems and nonlinear programs (for which explicit representations of the generalized Jacobians of the associated stationarity equations are available), such regularity assumptions are closely connected to strong second-order sufficient conditions. However, general characterizations are not well understood. In this paper, we investigate a strong second-order sufficient condition ($\mathrm{SSOSC}$) for composite problems whose nonsmooth part has a generalized conic-quadratic second subderivative. We discuss the relationship between the $\mathrm{SSOSC}$ and other second order-type conditions that involve the generalized Jacobians of the normal map. In particular, these two conditions are equivalent under certain structural assumptions on the generalized Jacobian matrix of the proximity operator. Leveraging second-order variational techniques and properties, we then verify that the introduced structural conditions hold for a broad class of $C^2$-strictly decomposable functions. Finally, it is shown that the $\mathrm{SSOSC}$ is further equivalent to the strong metric regularity of the subdifferential, the normal map, and the natural residual.