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In this article, we address a class of non convex, integer, non linear mathematical programs using dynamic programming. The mathematical program considered, whose properties are studied in this article, may be used to model the optimal…
Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic…
Sparse principal component analysis with global support (SPCAgs), is the problem of finding the top-$r$ leading principal components such that all these principal components are linear combinations of a common subset of at most $k$…
We present an exact Bayesian inference method for discrete statistical models, which can find exact solutions to a large class of discrete inference problems, even with infinite support and continuous priors. To express such models, we…
We introduce Supersparse Linear Integer Models (SLIM) as a tool to create scoring systems for binary classification. We derive theoretical bounds on the true risk of SLIM scoring systems, and present experimental results to show that SLIM…
We propose a flexible gradient-based framework for learning linear programs from optimal decisions. Linear programs are often specified by hand, using prior knowledge of relevant costs and constraints. In some applications, linear programs…
The 0-1 integer linear programming feasibility problem is an important NP-complete problem. This paper proposes a continuous-time dynamical system for solving that problem without getting trapped in non-solution local minima. First, the…
Designing computational experiments involving $\ell_1$ minimization with linear constraints in a finite-dimensional, real-valued space for receiving a sparse solution with a precise number $k$ of nonzero entries is, in general, difficult.…
Discrete integration is a fundamental problem in computer science that concerns the computation of discrete sums over exponentially large sets. Despite intense interest from researchers for over three decades, the design of scalable…
The challenge of mastering computational tasks of enormous size tends to frequently override questioning the quality of the numerical outcome in terms of accuracy. By this we do not mean the accuracy within the discrete setting, which…
Machine learning (ML) algorithms are predictively competitive algorithms with many human-impact applications. However, the issue of long execution time remains unsolved in the literature for high-dimensional spaces. This study proposes…
We propose an inexact variable-metric proximal point algorithm to accelerate gradient-based optimization algorithms. The proposed scheme, called QNing can be notably applied to incremental first-order methods such as the stochastic…
We study the complexity of approximating Wassertein barycenter of $m$ discrete measures, or histograms of size $n$ by contrasting two alternative approaches, both using entropic regularization. The first approach is based on the Iterative…
In the Subspace Clustering with Missing Data (SCMD) problem, we are given a collection of n partially observed d-dimensional vectors. The data points are assumed to be concentrated near a union of low-dimensional subspaces. The goal of SCMD…
Characterizing the ground-state properties of disordered systems, such as spin glasses and combinatorial optimization problems, is fundamental to science and engineering. However, computing exact ground states and counting their…
We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or non-negative integer…
Answer Set Programming (ASP) has emerged as a promising paradigm in knowledge representation and automated reasoning owing to its ability to model hard combinatorial problems from diverse domains in a natural way. Building on advances in…