Convex Optimization

A T-algebraic approach to primal-dual interior-point algorithms

  • Chek Beng Chua
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming

    Three primal-dual interior-point algorithms for homogeneous cone programming are presented. They are a short-step algorithm, a large-update algorithm, and a predictor-corrector algorithm. These algorithms are described and analyzed based on a characterization of homogeneous cone via T-algebra. The analysis show that the algorithms have polynomial iteration complexity. CitationDivision of Mathematical Sciences, Nanyang Technological University, December … Read more

    Exact regularization of convex programs

  • Michael P. Friedlander
  • Paul Tseng
    • Categories Convex Optimization Tags , , , , , , , ,

    The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a … Read more

    Linear convergence of a modified Frank-Wolfe algorithm for computing minimum volume ellipsoids

  • Selin Damla Ahipasaoglu
  • Peng Sun
  • Michael J. Todd
    • Categories Convex Optimization Tags , , ,

    We show the linear convergence of a simple first-order algorithm for the minimum-volume enclosing ellipsoid problem and its dual, the D-optimal design problem of statistics. Computational tests confirm the attractive features of this method. CitationOptimization Methods and Software 23 (2008), 5–19. ArticleDownload View PDF

    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

    Consistency of robust portfolio estimators

  • Katrin Schöttle
  • Ralf Werner
    • Categories Convex Optimization, Finance and Economics, Statistics Tags ,

    It is a matter of common knowledge that traditional Markowitz optimization based on sample means and covariances performs poorly in practice. For this reason, diverse attempts were made to improve performance of portfolio optimization. In this paper, we investigate three popular portfolio selection models built upon classical mean-variance theory. The first model is an extension … Read more

    PROXIMAL THRESHOLDING ALGORITHM FOR MINIMIZATION OVER ORTHONORMAL BASES

  • Patrick L. Combettes
  • Jean-Christophe Pesquet
    • Categories Convex Optimization, Infinite Dimensional Optimization, Unconstrained Optimization Tags , , ,

    The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing in particular several characterizations of … Read more

    An efficient method to compute traffic assignment problems with elastic demands

  • Frédéric Babonneau
  • Jean-Philippe Vial
    • Categories Convex Optimization, Transportation Tags , ,

    The traffic assignment problem with elastic demands can be formulated as an optimization problem, whose objective is sum of a congestion function and a disutility function. We propose to use a variant of the Analytic Center Cutting Plane Method to solve this problem. We test the method on instances with different congestion functions (linear with … Read more

    Computing Minimum Volume Enclosing Axis-Aligned Ellipsoids

  • Piyush Kumar
  • E. Alper Yildirim
    • Categories Convex Optimization, Nonlinear Optimization Tags , , ,

    Given a set of points $\cS = \{x^1,\ldots,x^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze an algorithm for the problem of computing a $(1 + \eps)$-approximation to the the minimum volume axis-aligned ellipsoid enclosing $\cS$. We establish that our algorithm is polynomial for fixed $\eps$. In addition, the algorithm returns a small … Read more

    Target following algorithms for semidefinite programming

  • Chek Beng Chua
    • Categories Convex Optimization, Semi-definite Programming

    We present a target-following framework for semidefinite programming, which generalizes the target-following framework for linear programming. We use this framework to build weighted path-following interior-point algorithms of three distinct flavors: short-step, predictor-corrector, and large-update. These algorithms have worse-case iteration bounds that parallel their counterparts in linear programming. We further consider the problem of finding analytic … Read more

    Mosco stability of proximal mappings in reflexive Banach spaces

  • Dan Butnariu
  • Elena Resmerita
    • Categories Convex Optimization Tags , , , , , ,

    In this paper we establish criteria for the stability of the proximal mapping \textrm{Prox} $_{\varphi }^{f}=(\partial \varphi +\partial f)^{-1}$ associated to the proper lower semicontinuous convex functions $\varphi $ and $f$ on a reflexive Banach space $X.$ We prove that, under certain conditions, if the convex functions $\varphi _{n}$ converge in the sense of Mosco … Read more