Convex Optimization

Generalized Self-Concordant Analysis of Frank-Wolfe algorithms

  • Pavel Dvurechensky
  • Shimrit Shtern
  • Kamil Safin
  • Mathias Staudigl
    • Categories Convex Optimization

    Projection-free optimization via different variants of the Frank-Wolfe (FW) method has become one of the cornerstones in large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant (GSC) functions do not necessarily feature a Lipschitz continuous gradient, nor are … Read more

    Largest small polygons: A sequential convex optimization approach

  • Christian Bingane
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Quadratic Programming Tags , , , , ,

    A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically constrained quadratic optimization problem. We propose to solve this problem with a … Read more

    On strong duality, theorems of the alternative, and projections in conic optimization

  • Temitayo Ajayi
  • Akshay Gupte
  • Amin Khademi
  • Andrew J. Schaefer
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , ,

    A conic program is the problem of optimizing a linear function over a closed convex cone intersected with an affine preimage of another cone. We analyse three constraint qualifications, namely a Closedness CQ, Slater CQ, and Boundedness CQ (also called Clark- Duffin theorem), that are sufficient for achieving strong duality and show that the first … Read more

    Dual optimal design and the Christoffel-Darboux polynomial

  • Yohann De Castro
  • Didier Henrion
  • Jean-Bernard Lasserre
  • Fabrice Gamboa
    • Categories Convex Optimization, Semi-definite Programming, Statistics Tags , , ,

    The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. Article Download View Dual optimal design and the Christoffel-Darboux polynomial

    Robust Convex Optimization: A New Perspective That Unifies And Extends

  • Dimitris Bertsimas
  • Dick den Hertog
  • Jean Pauphilet
  • Jianzhe Zhen
    • Categories Convex Optimization, Robust Optimization Tags , , ,

    Robust convex constraints are difficult to handle, since finding the worst-case scenario is equivalent to maximizing a convex function. In this paper, we propose a new approach to deal with such constraints that unifies approaches known in the literature and extends them in a significant way. The extension is either obtaining better solutions than the … Read more

    Online Convex Optimization Perspective for Learning from Dynamically Revealed Preferences

  • Violet (Xinying) Chen
  • Fatma Kılınç-Karzan
    • Categories Convex Optimization Tags , ,

    We study the problem of online learning (OL) from revealed preferences: a learner wishes to learn an agent’s private utility function through observing the agent’s utility-maximizing actions in a changing environment. We adopt an online inverse optimization setup, where the learner observes a stream of agent’s actions in an online fashion and the learning performance … Read more

    Learning Dynamical Systems with Side Information

  • Amir Ali Ahmadi
  • Bachir El Khadir
    • Categories Convex Optimization, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or … Read more

    A Modified Proximal Symmetric ADMM for Multi-Block Separable Convex Optimization with Linear Constraints

  • Yuan Shen
  • Xiayang Zhang
  • Yannian Zuo
    • Categories Convex Optimization Tags , , ,

    We consider the linearly constrained separable convex optimization problem whose objective function is separable w.r.t. $m$ blocks of variables. A bunch of methods have been proposed and well studied. Specifically, a modified strictly contractive Peaceman-Rachford splitting method (SC-PRCM) has been well studied in the literature for the special case of $m=3$. Based on the modified … Read more

    Optimization for Supervised Machine Learning: Randomized Algorithms for Data and Parameters

  • Filip Hanzely
    • Categories Convex Optimization, Stochastic Programming Tags , , , , ,

    Many key problems in machine learning and data science are routinely modeled as optimization problems and solved via optimization algorithms. With the increase of the volume of data and the size and complexity of the statistical models used to formulate these often ill-conditioned optimization tasks, there is a need for new efficient algorithms able to … Read more

    A geodesic interior-point method for linear optimization over symmetric cones

  • Frank Permenter
    • Categories Complementarity and Variational Inequalities, Convex Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    We develop a new interior-point method (IPM) for symmetric-cone optimization, a common generalization of linear, second-order-cone, and semidefinite programming. In contrast to classical IPMs, we update iterates with a geodesic of the cone instead of the kernel of the linear constraints. This approach yields a primal-dual-symmetric, scale-invariant, and line-search-free algorithm that uses just half the … Read more