Project data

Error Bounds, Critical Solutions and Numerical Methods for Smooth and Nonsmooth Optimization and Equilibrium Problems

Initiative: Trilateral Partnerships – Cooperation Projects between Scholars and Scientists from Ukraine, Russia and Germany
Allocation: Feb 10, 2016
Period of funding: 3 Years

Project information

An important issue of the project is the study of local error bounds and critical solutions of optimization and variational problems. Local Lipschitzian error bounds play the key role for the design and analysis of Newton-type methods with fast local convergence if the problem at hand has nonisolated solutions. Therefore, it is planend to extend the knowledge about conditions for the existence of error bounds for new problem classes. This will be a basis for the development of algorithmic techniques enabling local superlinear convergence in the case of nonisolated solutions. Research will rely on advanced mathematical tools, such as the theories of quadratic mappings and points of coincidence, which will also be developed within this project. Specifically, sufficient conditions for stability of the surjectivity property of a quadratic mapping will be studied, and then these results will be applied in order to derive inverse and implicit function theorems applicable at singular solutions of nonlinear equations. Moreover, it is expected to find conditions characterizing the structure of the set of coincidence points of two mapping. Along with Newton-type methods, it is also planned to study accelerated subgradient methods in the context of nonisolated solutions. First, the numerical behavior of existing modifications of r-algorithms and related subgradient methods for smooth and nonsmooth optimization problems with nonisolated solutions will be studied, including the development of reliable stopping criteria for such cases. Moreover, new subgradient methods with space transformation with provable geometric convergence (with respect to the localization volume) will be developed.

Project participants

  • Prof. Dr. Andreas Fischer

    Technische Universität Dresden
    Institut für Numerische Mathematik

  • Prof. Dr. Alexey Izmailov

    Lomonosov Moscow State University
    Faculty of Computational
    Mathematics and Cybernetics
    Department of Operations Research

  • Dr. Petro Stetsyuk

    National Academy of Sciences of Ukraine
    Glushkov Institute of Cybernetics
    Department of Nonsmooth Optimization Methods

  • Prof. Dr. Aram Arutyunov

    Peoples' Friendship University of Russia
    Faculty of Science
    Department of Nonlinear
    Analysis and Optimization