Andrés Gómez

A Parametric Approach for Solving Convex Quadratic Optimization with Indicators Over Trees

  • Aaresh Bhathena
  • Salar Fattahi
  • Andrés Gómez
  • Simge Küçükyavuz
    • Categories (Mixed) Integer Nonlinear Programming, Dynamic Programming, Quadratic Programming Tags , , , , ,

    This paper investigates convex quadratic optimization problems involving $n$ indicator variables, each associated with a continuous variable, particularly focusing on scenarios where the matrix $Q$ defining the quadratic term is positive definite and its sparsity pattern corresponds to the adjacency matrix of a tree graph. We introduce a graph-based dynamic programming algorithm that solves this … Read more

    Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators

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

    In this paper, we consider convex quadratic optimization problems with indicators on the continuous variables. In particular, we assume that the Hessian of the quadratic term is a Stieltjes matrix, which naturally appears in sparse graphical inference problems and others. We describe an explicit convex formulation for the problem by studying the Stieltjes polyhedron arising … Read more

    Robust support vector machines via conic optimization

  • Shaoning Han
  • Andrés Gómez
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Optimization in Data Science Tags , , , ,

    We consider the problem of learning support vector machines robust to uncertainty. It has been established in the literature that typical loss functions, including the hinge loss, are sensible to data perturbations and outliers, thus performing poorly in the setting considered. In contrast, using the 0-1 loss or a suitable non-convex approximation results in robust … Read more

    Learning Optimal Classification Trees Robust to Distribution Shifts

  • Nathan Justin
  • Sina Aghaei
  • Andrés Gómez
  • Phebe Vayanos
    • Categories (Mixed) Integer Linear Programming, Robust Optimization Tags , , , ,

    We consider the problem of learning classification trees that are robust to distribution shifts between training and testing/deployment data. This problem arises frequently in high stakes settings such as public health and social work where data is often collected using self-reported surveys which are highly sensitive to e.g., the framing of the questions, the time … Read more

    ODTlearn: A Package for Learning Optimal Decision Trees for Prediction and Prescription

  • Patrick Vossler
  • Sina Aghaei
  • Nathan Justin
  • Nathanael Jo
  • Andrés Gómez
  • Phebe Vayanos
    • Categories (Mixed) Integer Linear Programming, Optimization Software and Modeling Systems Tags , , , , , ,

    ODTLearn is an open-source Python package that provides methods for learning optimal decision trees for high-stakes predictive and prescriptive tasks based on the mixed-integer optimization (MIO) framework proposed in Aghaei et al. (2019) and several of its extensions. The current version of the package provides implementations for learning optimal classification trees, optimal fair classification trees, … Read more

    Solution Path of Time-varying Markov Random Fields with Discrete Regularization

  • Salar Fattahi
  • Andrés Gómez
    • Categories Dynamic Programming, Integer Programming, Optimization in Data Science

    We study the problem of inferring sparse time-varying Markov random fields (MRFs) with different discrete and temporal regularizations on the parameters. Due to the intractability of discrete regularization, most approaches for solving this problem rely on the so-called maximum-likelihood estimation (MLE) with relaxed regularization, which neither results in ideal statistical properties nor scale to the … Read more

    Stability-Adjusted 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 select the sparsity and robustness of linear regressors, techniques like k-fold cross-validation are commonly used for hyperparameter tuning. However, cross-validation substantially increases the … 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

    On polynomial time solvability of combinatorial Markov random fields

  • Shaoning Han
  • Andrés Gómez
  • Jong-Shi Pang
    • Categories (Mixed) Integer Nonlinear Programming, Data Science Applications

    The problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled as a mixed-integer program. This motivates us to study a general class of convex submodular optimization problems with indicator variables, which we show to be polynomially solvable in this paper. The key insight is that, possibly after … 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