riemannian manifolds

Constraint qualifications and strong global convergence properties of an augmented Lagrangian method on Riemannian manifolds

  • Roberto Andreani
  • Kelvin Rodrigues Couto
  • Orizon Pereira Ferreira
  • Gabriel Haeser
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

    In the past years, augmented Lagrangian methods have been successfully applied to several classes of non-convex optimization problems, inspiring new developments in both theory and practice. In this paper we bring most of these recent developments from nonlinear programming to the context of optimization on Riemannian manifolds, including equality and inequality constraints. Many research have … Read more

    Riemannian Interior Point Methods for Constrained Optimization on Manifolds

  • Zhijian Lai
  • Akiko Yoshise
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , , ,

    We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method (RIPM), is for solving Riemannian  constrained optimization problems. We establish its local superlinear and quadratic convergence  under the standard assumptions. Moreover, we show its global convergence when it is combined with … Read more

    Global convergence of Riemannian line search methods with a Zhang-Hager-type condition

  • Harry Oviedo
    • Categories Nonlinear Optimization Tags , , , ,

    In this paper, we analyze the global convergence of a general non–monotone line search method on Riemannian manifolds. For this end, we introduce some properties for the tangent search directions that guarantee the convergence, to a stationary point, of this family of optimization methods under appropriate assumptions. A modified version of the non–monotone line search … Read more

    An implementation of the steepest descent method using retractions on riemannian manifolds

  • Ever Cruzado
  • Erik Alex Papa Quiroz
    • Categories Global Optimization Tags , , ,

    In 2008 Absil et al. published a book with optimization methods in Riemannian manifolds. The authors developed steepest descent, Newton, trust-region and conjugate gradients methods using an approximation of the geodesic called retraction. In this paper we present implementations of the of steepest descent method of Absil et al. using Matlab software. We show the … Read more

    DIRECT SEARCH ALGORITHMS OVER RIEMANNIAN MANIFOLDS

  • David W. Dreisigmeyer
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    We generalize the Nelder-Mead simplex and LTMADS algorithms and, the frame based methods for function minimization to Riemannian manifolds. Examples are given for functions defined on the special orthogonal Lie group $\mathcal{SO}(n)$ and the Grassmann manifold $\mathcal{G}(n,k)$. Our main examples are applying the generalized LTMADS algorithm to equality constrained optimization problems and, to the Whitney … Read more

    EQUALITY CONSTRAINTS, RIEMANNIAN MANIFOLDS AND DIRECT SEARCH METHODS

  • David W. Dreisigmeyer
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We present a general procedure for handling equality constraints in optimization problems that is of particular use in direct search methods. First we will provide the necessary background in differential geometry. In particular, we will see what a Riemannian manifold is, what a tangent space is, how to move over a manifold and how to … Read more

    Dini Derivative and a Characterization for Lipschitz and Convex Functions on Riemannian Manifolds

  • Orizon Pereira Ferreira
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    Dini derivative on Riemannian manifold setting is studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given. ArticleDownload View PDF