Abstract：

This talk presents two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are either of class C1;1, or are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. The proposed globally convergent algorithms are of two types. The first one extends the damped Newton method and requires positive-definiteness of the generalized Hessians for its well-posedness and effcient performance, while the other algorithm is of the regularized Newton type being well-defined when the generalized Hessians are merely positive-semidefinite. The obtained convergence rates for both methods are at least linear, but becomes superlinear under the semismooth property of subgradient mappings. Problems of convex composite optimization are investigated with and without the strong convexity assumption on smooth parts of objective functions by implementing the machinery of forward-backward envelopes. Numerical experiments are conducted for Lasso problems and for box constrained quadratic programs with providing performance comparisons of the new algorithms and some other first-order and second-order methods that are highly recognized in nonsmooth optimization.

About the speaker：

Boris Mordukhovich is an American mathematician recognized for his research in the areas of nonlinear analysis, optimization, and control theory. Mordukhovich is an AMS Fellow of the Inaugural Class, a SIAM Fellow, and a recipient of many international awards and honors. Currently he is Distinguished University Professor and Lifetime Scholar of the Academy of Scholars at Wayne State University (Vice President, 2009–2010 and President, 2010–2011). He developed constructions of generalized differentiation (bearing now his name), and their development and applications to classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields.