Prof. Boris S. Mordukhovich

Wayne State University, Detroit, USA

Variational Analysis in Numerical Optimization

Abstract. This paper proposes and justifies two new 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 efficient 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 so-called semismooth_ property of subgradient mappings. Problems of convex composite optimization are investigated with and without the strong convexity assumption on of smooth parts of objective functions by implementing the machinery of forward-backward envelopes. Numerical experiments are conducted for a basic class of Lasso problems by providing performance comparisons of the new algorithms with some other first-order and second-order methods that are highly recognized in nonsmooth optimization. This talk is based on the joint work with P. D. Khanh, T. V. Phat and D. B. Tran.

Plenary Speakers

Prof. Roman A. Polyak
George Mason University, USA
Prof. Srinivas Chakravarthy
Kettering University, Flint, USA
Prof. Arkadii Chikrii
V. M. Glushkov Institute of Cybernetics, Kyiv, Ukraine
Prof. Boris S. Mordukhovich
Wayne State University, Detroit, USA
Prof. Janusz Kacprzyk
Warsaw University of Technology, Poland
Prof. Yedilkhan Amirgaliyev
Institute of Information and Computing Technologies, Almaty, Kazakhstan
Prof. Adil Bagirov
Centre for Smart Analytics, Institute of Innovation, Science and Sustainability, Federation University Australia, Australia
Prof. Rza Bashirov
Eastern Mediterranean University, Turkey
Prof. Refail Kasimbeyli
Eskişehir Technical University, Türkiye
Prof. Agasi Melikov
Institute of Control Systems, Ministry of Science and Education, Azerbaijan
Prof. Mais Farkhadov
Institute of Control Sciences of RAS, Russia