This paper concerns with the group zero-norm regularized least squares estimator which, in terms of the variational characterization of the zero-norm, can be obtained from a mathematical program with equilibrium constraints (MPEC). By developing the global exact penalty for the MPEC, this estimator is shown to arise from an exact penalization problem that not only … Read more
We consider a bound for the maximum-entropy sampling problem (MESP) that is based on solving a max-det problem over a relaxation of the Boolean Quadric Polytope (BQP). This approach to MESP was first suggested by Christoph Helmberg over 15 years ago, but has apparently never been further elaborated or computationally investigated. We find that the … Read more
We propose a forecasting approach for solar flares based on data from Solar Cycle 24, taken by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) mission. In particular, we use the Space-weather HMI Active Region Patches (SHARP) product that facilitates cut-out magnetograms of solar active regions (AR) in the Sun … Read more
We address the problem of prescribing an optimal decision in a framework where its cost depends on uncertain problem parameters $Y$ that need to be learned from data. Earlier work by Bertsimas and Kallus (2014) transforms classical machine learning methods that merely predict $Y$ from supervised training data $[(x_1, y_1), \dots, (x_n, y_n)]$ into prescriptive … Read more
We study a class of stochastic programs where some of the elements in the objective function are random, and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two common … Read more
We study a class of quadratic stochastic programs where the distribution of random variables has unknown parameters. A traditional approach is to estimate the parameters using a maximum likelihood estimator (MLE) and to use this as input in the optimization problem. For the unconstrained case, we show that an estimator that “shrinks” the MLE towards … Read more
The bootstrap is a nonparametric approach for calculating quantities, such as confidence intervals, directly from data. Since calculating exact bootstrap quantities is believed to be intractable, randomized resampling algorithms are traditionally used. Motivated by the fact that the variability from randomization can lead to inaccurate outputs, we propose a deterministic approach. First, we establish several … Read more
Testing a given matrix for membership in the family of Bernoulli matrices is a longstanding problem, the many applications of Bernoulli vectors in computer science, finance, medicine, and operations research emphasize its practical relevance. A novel approach towards this problem was taken by [Fiebig et al., 2017] for lowdimensional settings d
Nonconvex penalty methods for sparse modeling in linear regression have been a topic of fervent interest in recent years. Herein, we study a family of nonconvex penalty functions that we call the trimmed Lasso and that offers exact control over the desired level of sparsity of estimators. We analyze its structural properties and in doing … Read more
The general formulation for finding the L1-norm best-fit subspace for a point set in $m$-dimensions is a nonlinear, nonconvex, nonsmooth optimization problem. In this paper we present a procedure to estimate the L1-norm best-fit one-dimensional subspace (a line through the origin) to data in $\Re^m$ based on an optimization criterion involving linear programming but which … Read more