James V. Burke

Algorithms for Block Tridiagonal Systems: Foundations and New Results for Generalized Kalman Smoothing

  • Aleksandr Y. Aravkin
  • James V. Burke
  • Bradley Bell
  • Gianluigi Pillonetto
    • Categories Control Applications, Convex Optimization

    Block tridiagonal systems appear in classic Kalman smoothing problems, as well in generalized Kalman smoothing, where problems may have nonsmooth terms, singular covariance, constraints, nonlinear models, and unknown parameters. In this paper, first we interpret all the classic smoothing algorithms as different approaches to solve positive definite block tridiagonal linear systems. Then, we obtain new … Read more

    Fast Robust Methods for Singular State-Space Models

  • Aleksandr Y. Aravkin
  • James V. Burke
  • Sarah Webster
  • Jonathan Jonker
  • Gianluigi Pillonetto
    • Categories Control Applications, Nonsmooth Optimization, Statistics Tags , ,

    State-space models are used in a wide range of time series analysis applications. Kalman filtering and smoothing are work-horse algorithms in these settings. While classic algorithms assume Gaussian errors to simplify estimation, recent advances use a broad range of optimization formulations to allow outlier-robust estimation, as well as constraints to capture prior information. Here we … Read more

    Gradient Sampling Methods for Nonsmooth Optimization

  • James V. Burke
  • Frank E. Curtis
  • Adrian Lewis
  • Michael L. Overton
  • Lucas E. A. Simões
    • Categories Nonsmooth Optimization

    This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various … Read more

    A Dynamic Penalty Parameter Updating Strategy for Matrix-Free Sequential Quadratic Optimization

  • James V. Burke
  • Frank E. Curtis
  • Jiashan Wang
  • Hao Wang
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    This paper focuses on the design of sequential quadratic optimization (commonly known as SQP) methods for solving large-scale nonlinear optimization problems. The most computationally demanding aspect of such an approach is the computation of the search direction during each iteration, for which we consider the use of matrix-free methods. In particular, we develop a method … Read more

    Subdifferentiation and Smoothing of Nonsmooth Integral Functionals

  • James V. Burke
  • Xiaojun Chen
  • Hailin Sun
    • Categories Nonsmooth Optimization Tags , , ,

    The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about calculation of Clarke subgradients for the … Read more

    CONVEX GEOMETRY OF THE GENERALIZED MATRIX-FRACTIONAL FUNCTION

  • James V. Burke
  • Tim Hoheisel
  • Yuan Gao
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    Generalized matrix-fractional (GMF) functions are a class of matrix support functions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix optimization problems associated with inverse problems, regularization and learning. In this paper we dramatically simplify the support function representation for GMF functions as well as the representation of … Read more

    Foundations of gauge and perspective duality

  • Aleksandr Y. Aravkin
  • James V. Burke
  • Dmitriy Drusvyatskiy
  • Michael P. Friedlander
  • Kellie J. MacPhee
    • Categories Convex Optimization Tags , , ,

    Common numerical methods for constrained convex optimization are predicated on efficiently computing nearest points to the feasible region. The presence of a design matrix in the constraints yields feasible regions with more complex geometries. When the functional components are gauges, there is an equivalent optimization problem—the gauge dual– where the matrix appears only in the … Read more

    Level-set methods for convex optimization

  • Aleksandr Y. Aravkin
  • James V. Burke
  • Dmitriy Drusvyatskiy
  • Michael P. Friedlander
  • Scott Roy
    • Categories Convex Optimization Tags , , , ,

    Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the objective and constraint functions, and instead approximately solves a sequence of parametric level-set problems. A zero-finding procedure, based on inexact function evaluations and … Read more

    On a new class of matrix support functionals with applications

  • James V. Burke
  • Tim Hoheisel
    • Categories Convex and Nonsmooth Optimization, Quadratic Programming, Semi-definite Programming Tags , , , , , ,

    A new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, and the nuclear norm. The support function is based on the graph of the product of a matrix with its transpose. Closed form expressions for the support … Read more

    Iterative Reweighted Linear Least Squares for Exact Penalty Subproblems on Product Sets

  • James V. Burke
  • Frank E. Curtis
  • Hao Wang
  • Jiashan Wang
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonlinear Optimization Tags , , , , ,

    We present two matrix-free methods for solving exact penalty subproblems on product sets that arise when solving large-scale optimization problems. The first approach is a novel iterative reweighting algorithm (IRWA), which iteratively minimizes quadratic models of relaxed subproblems while automatically updating a relaxation vector. The second approach is based on alternating direction augmented Lagrangian (ADAL) … Read more