Convex and Nonsmooth Optimization

Primal-dual first-order methods with ${\cal O}(1/\epsilon)$ iteration-complexity for cone programming

  • Guanghui Lan
  • Zhaosong Lu
  • Renato D.C. Monteiro
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , ,

    In this paper we consider the general cone programming problem, and propose primal-dual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method \cite{Nest83-1,Nest05-1}, Nesterov’s smooth approximation scheme \cite{Nest05-1}, and Nemirovski’s prox-method \cite{Nem05-1}, and propose a variant of Nesterov’s optimal method which has … Read more

    A PARALLEL interior point decomposition algorithm for block-angular semidefinite programs

  • Kartik Krishnan Sivaramakrishnan
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , ,

    We present a two phase interior point decomposition framework for solving semidefinite (SDP) relaxations of sparse maxcut, stable set, and box constrained quadratic programs. In phase 1, we suitably modify the {\em matrix completion} scheme of Fukuda et al. \cite{fukuda_et_al} to preprocess an existing SDP into an equivalent SDP in the block-angular form. In phase … Read more

    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

    Selective Gram-Schmidt orthonormalization for conic cutting surface algorithms

  • Vasile L. Basescu
  • John E. Mitchell
    • Categories Linear, Cone and Semidefinite Programming, Nonsmooth Optimization Tags , , ,

    It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly … 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

    A Proximal Point Algorithm with Bregman Distances for Quasiconvex Optimization over the Positive Orthant

  • João Xavier da Cruz Neto
  • Paulo Roberto Oliveira
  • Antoine Soubeyran
  • Sissy da S. Souza
    • Categories Generalized Convexity/Monoticity Tags , , ,

    We present an interior proximal point method with Bregman distance, whose Bregman function is separable and the zone is the interior of the positive orthant, for solving the quasiconvex optimization problem under nonnegative constraints. We establish the well-definedness of the sequence generated by our algorithm and we prove convergence to a solution point when the … 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

    Simplex-type algorithm for optimizing a pseudolinear quadratic fractional function over a polytope

  • Miklós Ujvári
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity

    Recently Cambini and Carosi described a characterization of pseudolinearity of quadratic fractional functions. A reformulation of their result was given by Rapcsák. Using this reformulation, in this paper we describe an alternative proof of the Cambini–Carosi Theorem. Our proof is shorter than the proof given by Cambini–Carosi and less involved than the proof given by … Read more