Andrés Gómez

Optimal Cross-Validation for Sparse Linear Regression

  • Ryan Cory-Wright
  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming, Data Science Algorithms

    Given a high-dimensional covariate matrix and a response vector, ridge-regularized sparse linear regression selects a subset of features that explains the relationship between covariates and the response in an interpretable manner. To choose hyperparameters that control the sparsity level and amount of regularization, practitioners commonly use k-fold cross-validation. However, cross-validation substantially increases the computational cost … Read more

    Outlier detection in regression: conic quadratic formulations

  • Andrés Gómez
  • José Neto
    • Categories (Mixed) Integer Nonlinear Programming, Statistics Tags , , ,

    In many applications, when building linear regression models, it is important to account for the presence of outliers, i.e., corrupted input data points. Such problems can be formulated as mixed-integer optimization problems involving cubic terms, each given by the product of a binary variable and a quadratic term of the continuous variables. Existing approaches in … Read more

    A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation

  • Andrés Gómez
  • Weijun Xie
    • Categories (Mixed) Integer Nonlinear Programming

    In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we … Read more

    On the convex hull of convex quadratic optimization problems with indicators

  • Linchuan Wei
  • Alper Atamturk
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming Tags , ,

    We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of … Read more

    Compact extended formulations for low-rank functions with indicator variables

  • Shaoning Han
  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming Tags , , ,

    We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions Extended formulations describing the convex hull of such … Read more

    A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

  • Peijing Liu
  • Salar Fattahi
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Statistics Tags , , , , , ,

    In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This … Read more

    Comparing Solution Paths of Sparse Quadratic Minimization with a Stieltjes Matrix

  • Ziyu He
  • Shaoning Han
  • Andrés Gómez
  • Ying Cui
  • Jong-Shi Pang
    • Categories Nonsmooth Optimization, Quadratic Programming, Statistics Tags , , ,

    This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “L0-path” where the discontinuous L0-function provides the exact sparsity count; the “L1-path” where the L1-function provides a convex surrogate of sparsity count; … Read more

    Learning Optimal Prescriptive Trees from Observational Data

  • Nathanael Jo
  • Sina Aghaei
  • Andrés Gómez
  • Phebe Vayanos
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences Tags , , , ,

    We consider the problem of learning an optimal prescriptive tree (i.e., an interpretable treatment assignment policy in the form of a binary tree) of moderate depth, from observational data. This problem arises in numerous socially important domains such as public health and personalized medicine, where interpretable and data-driven interventions are sought based on data gathered … Read more

    Single-neuron convexifications for binarized neural networks

  • Andrés Gómez
  • Shaoning Han
    • Categories (Mixed) Integer Linear Programming, Cutting Plane Approaches, Statistics Tags , , , ,

    Binarized neural networks are an important class of neural network in deep learning due to their computational efficiency. This paper contributes towards a better understanding of the structure of binarized neural networks, specifically, ideal convex representations of the activation functions used. We describe the convex hull of the graph of the signum activation function associated … Read more

    Scalable Inference of Sparsely-changing Markov Random Fields with Strong Statistical Guarantees

  • Salar Fattahi
  • Andrés Gómez
    • Categories Combinatorial Optimization, Statistics Tags , ,

    In this paper, we study the problem of inferring time-varying Markov random fields (MRF), where the underlying graphical model is both sparse and changes sparsely over time. Most of the existing methods for the inference of time-varying MRFs rely on the regularized maximum likelihood estimation (MLE), that typically suffer from weak statistical guarantees and high … Read more