Hongbo Dong

Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

  • Hongbo Dong
  • Yunqi Luo
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Second-Order Cone Programming

    Decades of advances in mixed-integer linear programming (MILP) and recent development in mixed-integer second-order-cone programming (MISOCP) have translated very mildly to progresses in global solving nonconvex mixed-integer quadratically constrained programs (MIQCP). In this paper we propose a new approach, namely Compact Disjunctive Approximation (CDA), to approximate nonconvex MIQCP to arbitrary precision by convex MIQCPs, which … Read more

    On the Linear Convergence of Difference-of-convex Algorithms for Nonsmooth DC Programming

  • Hongbo Dong
  • Tao Min
    • Categories Nonsmooth Optimization Tags , , ,

    In this paper we consider the linear convergence of algorithms for minimizing difference- of-convex functions with convex constraints. We allow nonsmoothness in both of the convex and concave components in the objective function, with a finite max structure in the concave compo- nent. Our focus is on algorithms that compute (weak and standard) d(irectional)-stationary points … Read more

    On Integer and MPCC Representability of Affine Sparsity

  • Hongbo Dong
    • Categories (Mixed) Integer Nonlinear Programming, Complementarity and Variational Inequalities Tags , ,

    In addition to sparsity, practitioners of statistics and machine learning often wish to promote additional structures in their variable selection process to incorporate prior knowledge. Borrowing the modeling power of linear systems with binary variables, many of such structures can be faithfully modeled as so-called affine sparsity constraints (ASC). In this note we study conditions … Read more

    Structural Properties of Affine Sparsity Constraints

  • Miju Ahn
  • Hongbo Dong
  • Jong-Shi Pang
    • Categories Convex and Nonsmooth Optimization, Statistics

    We introduce a new constraint system for sparse variable selection in statistical learning. Such a system arises when there are logical conditions on the sparsity of certain unknown model parameters that need to be incorporated into their selection process. Formally, extending a cardinality constraint, an affine sparsity constraint (ASC) is defined by a linear inequality … Read more

    Faster Estimation of High-Dimensional Vine Copulas with Automatic Differentiation

  • Hongbo Dong
  • Haijun Li
  • Wei Li
    • Categories Statistics

    Vine copula is an important tool in modeling dependence structures of continuous-valued random variables. The maximum likelihood estimation (MLE) for vine copulas has long been considered computationally difficult in higher dimensions, even in 10 or 20 dimensions. Current computational practice, including the implementation in the state-of- the-art R package VineCopula, suffers from the bottleneck of … Read more

    Regularization vs. Relaxation: A convexification perspective of statistical variable selection

  • Kun Chen
  • Hongbo Dong
  • Jeff Linderoth
    • Categories (Mixed) Integer Nonlinear Programming, Linear, Cone and Semidefinite Programming, Statistics Tags , , ,

    Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an $\ell_0$-norm penalty. Exactly enforcing the $\ell_0$-norm penalty is computationally intractable for larger scale problems, so different sparsity-inducing penalty … Read more

    Relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations

  • Hongbo Dong
    • Categories Global Optimization Theory, Quadratic Programming, Semi-definite Programming

    The current bottleneck of globally solving mixed-integer (nonconvex) quadratically constrained problem (MIQCP) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We pro- pose a cutting surface procedure based on multiple diagonal perturbations to derive strong convex quadratic relaxations for nonconvex quadratic problem with separable constraints. Our … Read more

    On valid inequalities for quadratic programming with continuous variables and binary indicators

  • Hongbo Dong
  • Jeff Linderoth
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Theory, Quadratic Programming Tags ,

    In this paper we study valid inequalities for a fundamental set that involves a continuous vector variable x in [0,1]^n, its associated quadratic form x x’ and its binary indicators. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). We treat valid inequalities for this set as lifted from QPB, which … Read more

    Representing quadratically constrained quadratic programs as generalized copositive programs

  • Samuel Burer
  • Hongbo Dong
    • Categories Global Optimization Theory, Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , ,

    We show that any nonconvex quadratically constrained quadratic program(QCQP) can be represented as a generalized copositive program. In fact,we provide two representations. The first is based on the concept of completely positive (CP) matrices over second order cones, while the second is based on CP matrices over the positive semidefinte cone. Our analysis assumes that … Read more

    Symmetric tensor approximation hierarchies for the completely positive cone

  • Hongbo Dong
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    In this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors. We show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone, and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone. As … Read more