Alexey F. Izmailov

Behavior of Newton-type methods near critical solutions of nonlinear equations with semismooth derivatives

  • Andreas Fischer
  • Alexey F. Izmailov
  • Mario Jelitte
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , ,

    Having in mind singular solutions of smooth reformulations of complementarity problems, arising unavoidably when the solution in question violates strict complementarity, we study the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming that the operator of the equation possesses a strongly semismooth derivative, but is not necessarily twice differentiable. These smoothness restrictions … Read more

    ACCELERATING CONVERGENCE OF A GLOBALIZED SEQUENTIAL QUADRATIC PROGRAMMING METHOD TO CRITICAL LAGRANGE MULTIPLIERS

  • Alexey F. Izmailov
    • Categories Constrained Nonlinear Optimization

    This paper concerns the issue of asymptotic acceptance of the true Hessian and the full step by the sequential quadratic programming algorithm for equality-constrained optimization problems. In order to enforce global convergence, the algorithm is equipped with a standard Armijo linesearch procedure for a nonsmooth exact penalty function. The specificity of considerations here is that … Read more

    Continuous selections of solutions for locally Lipschitzian equations

  • A.V. Arutyunov
  • Alexey F. Izmailov
  • S.E. Zhukovskiy
    • Categories Complementarity and Variational Inequalities, Nonlinear Systems and Least-Squares, Nonsmooth Optimization

    This paper answers in affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given … Read more

    Local attractors of newton-type methods for constrained equations and complementarity problems with nonisolated solutions

  • Andreas Fischer
  • Alexey F. Izmailov
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Nonlinear Systems and Least-Squares

    For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well-de ned … Read more

    Convergence Conditions for Newton-type Methods Applied to Complementarity Systems with Nonisolated Solutions

  • Andreas Fischer
  • Alexey F. Izmailov
  • Markus Herrich
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization Tags , , , , , ,

    We consider a class of Newton-type methods for constrained systems of equations that involve complementarity conditions. In particular, at issue are the constrained Levenberg–Marquardt method and the recently introduced Linear Programming-Newton method, designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We … Read more

    Local Convergence of the Method of Multipliers for Variational and Optimization Problems under the Sole Noncriticality Assumption

  • Alexey F. Izmailov
  • A. S. Kurennoy
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    We present local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local superlinear convergence is established under the … Read more

    Abstract Newtonian Frameworks and Their Applications

  • Alexey F. Izmailov
  • A. S. Kurennoy
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , , ,

    We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving … Read more

    Attraction of Newton method to critical Lagrange multipliers: fully quadratic case

  • Alexey F. Izmailov
  • E.I. Uskov
    • Categories Constrained Nonlinear Optimization Tags , , ,

    All previously known results concerned with attraction of Newton-type iterations for optimality systems to critical Lagrange multipliers were a posteriori by nature: they were showing that in case of convergence, the dual limit is in a sense unlikely to be noncritical. This paper suggests the first a priori result in this direction, showing that critical … Read more

    STRONGLY REGULAR NONSMOOTH GENERALIZED EQUATIONS (REVISED)

  • Alexey F. Izmailov
    • Categories Complementarity and Variational Inequalities Tags , , , , ,

    This note suggests the implicit function theorem for generalized equations, unifying Robinson’s theorem for strongly regular generalized equations and Clarke’s implicit function theorem for equations with Lipschitz-continuous mappings. ArticleDownload View PDF

    ON REGULARITY CONDITIONS FOR COMPLEMENTARITY PROBLEMS

  • Alexey F. Izmailov
  • A. S. Kurennoy
    • Categories Complementarity and Variational Inequalities Tags , , , , , , , , , ,

    In the context of mixed complementarity problems, various concepts of solution regularity are known, each of them playing a certain role in related theoretical and algorithmic developments. In this note, we provide the complete picture of relations between the most important regularity conditions for mixed complementarity problems. A special attention is paid to the particular … Read more