Variational Properties of Decomposable Functions. Part II: Strong Second-Order Theory
Wenqing Ouyang, Andre Milzarek
TL;DR
The paper develops a strong second-order sufficient condition (SSOSC) for composite nonsmooth optimization problems where the nonsmooth part has a generalized conic-quadratic second subderivative. It builds a bridge from SSOSC to a Jacobian-based second-order condition via structured assumptions on the generalized Jacobian of the proximity operator, and shows that these conditions yield uniform invertibility essential for semismooth Newton convergence. The authors demonstrate that the structural assumptions hold for a broad class of C^2-strictly decomposable functions under a mild OTP geometry on the constraint set, and prove equivalences between SSOSC and strong metric regularity for the subdifferential, normal map, and natural residual. Collectively, these results provide robust theoretical guarantees for local convergence of semismooth Newton methods in conic-type composite problems without requiring strict complementarity, with implications for nonlinear programs and related variational problems.
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.