generalized equation

Computation of Graphical Derivative for a Class of Normal Cone Mappings under a Very Weak Condition

  • Huy Chieu Nguyen
  • Le Van Hien
    • Categories Nonsmooth Optimization Tags , , , ,

    Let $\Gamma:=\{x\in \R^n\, |\, q(x)\in\Theta\},$ where $q: \R^n\rightarrow\R^m$ is a twice continuously differentiable mapping, and $\Theta$ is a nonempty polyhedral convex set in $\R^m.$ In this paper, we first establish a formula for exactly computing the graphical derivative of the normal cone mapping $N_\Gamma:\R^n\rightrightarrows\R^n,$ $x\mapsto N_\Gamma(x),$ under the condition that $M_q(x):=q(x)-\Theta$ is metrically subregular at … Read more

    Approximations and Generalized Newton Methods

  • Diethard Klatte
  • Bernd Kummer
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , , ,

    We study local convergence of generalized Newton methods for both equations and inclusions by using known and new approximations and regularity properties at the solution. Including Kantorovich-type settings, our goal are statements about all (not only some) Newton sequences with appropriate initial points. Our basic tools are results of Klatte-Kummer (2002) and Kummer (1988, 1995), … Read more

    Directional H”older metric subregularity and application to tangent cones

  • Nguyen Huu Tron
  • Huynh Van Ngai
  • Phan Nhat Tinh
    • Categories Convex and Nonsmooth Optimization Tags , , , , ,

    In this work, we study directional versions of the H\”olderian/Lipschitzian metric subregularity of multifunctions. Firstly, we establish variational characterizations of the H\”olderian/Lipschitzian directional metric subregularity by means of the strong slopes and next of mixed tangency-coderivative objects . By product, we give second-order conditions for the directional Lipschitzian metric subregularity and for the directional metric … 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

    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. Article Download View STRONGLY REGULAR NONSMOOTH GENERALIZED EQUATIONS (REVISED)