Globally convergent coderivative-based generalized Newton methods in nonsmooth optimization

This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained constrained problems of nonsmooth optimization by using tools variational analysis generalized differentiation. Both are coderivative-based employ Hessians (coderivatives subgradient mappings) associated with objective functions, which either class $${{\mathcal {C}}}^{1,1}$$ , or represented in the form convex composite optimization, where one terms may be extended-real-valued. The proposed algorithms types. first extends damped Newton method requires positive-definiteness for its well-posedness efficient performance, while other algorithm is regularized being well-defined when merely positive-semidefinite. obtained convergence rates both at least linear, but become superlinear under semismooth $$^*$$ property mappings. Problems investigated without strong convexity assumption on smooth parts functions implementing machinery forward–backward envelopes. Numerical experiments conducted Lasso box quadratic programs providing performance comparisons new some first-order second-order that highly recognized optimization.