Convex Optimization

Primal-dual relationship between Levenberg-Marquardt and central trajectories for linearly constrained convex optimization

  • Roger Behling
  • Clóvis Caesar Gonzaga
  • Gabriel Haeser
    • Categories Convex Optimization, Quadratic Programming Tags , , , , , ,

    We consider the minimization of a convex function on a compact polyhedron defined by linear equality constraints and nonnegative variables. We define the Levenberg-Marquardt (L-M) and central trajectories starting at the analytic center and using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of … Read more

    Mean squared error minimization for inverse moment problems

  • Didier Henrion
  • Jean-Bernard Lasserre
  • Martin Mevissen
    • Categories Convex Optimization, Semi-definite Programming

    We consider the problem of approximating the unknown density $u\in L^2(\Omega,\lambda)$ of a measure $\mu$ on $\Omega\subset\R^n$, absolutely continuous with respect to some given reference measure $\lambda$, from the only knowledge of finitely many moments of $\mu$. Given $d\in\N$ and moments of order $d$, we provide a polynomial $p_d$ which minimizes the mean square error … Read more

    Fast global convergence of gradient methods for high-dimensional statistical recovery

  • Alekh Agarwal
  • Martin J Wainwright
  • Sahand Negahban
    • Categories Convex Optimization, Statistics Tags , ,

    Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension $\pdim$ to grow with (and possibly exceed) … Read more

    On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers

  • Wei Deng
  • Wotao Yin
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , ,

    The formulation min f(x)+g(y) subject to Ax+By=b, where f and g are extended-value convex functions, arises in many application areas such as signal processing, imaging and image processing, statistics, and machine learning either naturally or after variable splitting. In many common problems, one of the two objective functions is strictly convex and has Lipschitz continuous … Read more

    A Newton’s method for the continuous quadratic knapsack problem

  • Roberto Cominetti
  • Walter F. Mascarenhas
  • Paulo J. S. Silva
    • Categories Convex Optimization, Quadratic Programming Tags , , ,

    We introduce a new efficient method to solve the continuous quadratic knapsack problem. This is a highly structured quadratic program that appears in different contexts. The method converges after O(n) iterations with overall arithmetic complexity O(n²). Numerical experiments show that in practice the method converges in a small number of iterations with overall linear complexity, … Read more

    Learning Circulant Sensing Kernels

  • Stanley Osher
  • Yangyang Xu
  • Wotao Yin
    • Categories Convex Optimization, Data-Mining, Quadratic Programming Tags , , , , ,

    In signal acquisition, Toeplitz and circulant matrices are widely used as sensing operators. They correspond to discrete convolutions and are easily or even naturally realized in various applications. For compressive sensing, recent work has used random Toeplitz and circulant sensing matrices and proved their efficiency in theory, by computer simulations, as well as through physical … Read more

    An Efficient Augmented Lagrangian Method with Applications to Total Variation Minimization

  • Hong Jiang
  • Wotao Yin
  • Yin Zhang
  • Chengbo Li
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , ,

    Based on the classic augmented Lagrangian multiplier method, we propose, analyze and test an algorithm for solving a class of equality-constrained non-smooth optimization problems (chiefly but not necessarily convex programs) with a particular structure. The algorithm effectively combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each … Read more

    Convex computation of the region of attraction of polynomial control systems

  • Didier Henrion
  • Milan Korda
    • Categories Control Applications, Convex Optimization, Semi-definite Programming Tags , , , , , ,

    We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving a convex linear programming (LP) problem over the … Read more

    Bounds on Eigenvalues of Matrices Arising from Interior-Point Methods

  • Chen Greif
  • Erin Moulding
  • Dominique Orban
    • Categories Convex Optimization, Quadratic Programming Tags , , , , , , ,

    Interior-point methods feature prominently among numerical methods for inequality-constrained optimization problems, and involve the need to solve a sequence of linear systems that typically become increasingly ill-conditioned with the iterations. To solve these systems, whose original form has a nonsymmetric 3×3 block structure, it is common practice to perform block Gaussian elimination and either solve … Read more

    An adaptive accelerated first-order method for convex optimization

  • Renato D.C. Monteiro
  • Camilo Ortiz
  • Benar Fux Svaiter
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , , ,

    This paper presents a new accelerated variant of Nesterov’s method for solving composite convex optimization problems in which certain acceleration parameters are adaptively (and aggressively) chosen so as to substantially improve its practical performance compared to existing accelerated variants while at the same time preserve the optimal iteration-complexity shared by these methods. Computational results are … Read more