Related papers: Systems of Linear Equations over $\mathbb{F}_2$ an…
In the Max $r$-SAT problem, the input is a CNF formula with $n$ variables where each clause is a disjunction of at most $r$ literals. The objective is to compute an assignment which satisfies as many of the clauses as possible. While there…
We propose a general algorithm of constructing an extended formulation for any given set of linear constraints with integer coefficients. Our algorithm consists of two phases: first construct a decision diagram $(V,E)$ that somehow…
Solving a linear system $Ax=b$ is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are…
In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so…
We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic…
In this note we show that unsatisfiable systems of linear equations with a constant number of variables per equation over prime finite fields have polynomial-size constant-degree semi-algebraic proofs of unsatisfiability. These are proofs…
We present two matrix-free methods for approximately solving exact penalty subproblems that arise when solving large-scale optimization problems. The first approach is a novel iterative re-weighting algorithm (IRWA), which iteratively…
Incorporating a non-Euclidean variable metric to first-order algorithms is known to bring enhancement. However, due to the lack of an optimal choice, such an enhancement appears significantly underestimated. In this work, we establish a…
For many constraint satisfaction problems, the algorithm which chooses a random assignment achieves the best possible approximation ratio. For instance, a simple random assignment for {\sc Max-E3-Sat} allows 7/8-approximation and for every…
The Kaczmarz method for solving a linear system $Ax = b$ interprets such a system as a collection of equations $\left\langle a_i, x\right\rangle = b_i$, where $a_i$ is the $i-$th row of $A$, then picks such an equation and corrects $x_{k+1}…
Let $L$ be a set of $n$ lines in the plane, not necessarily in general position. We present an efficient algorithm for finding all the vertices of the arrangement $A(L)$ of maximum level, where the level of a vertex $v$ is the number of…
We consider the allocation of $m$ balls into $n$ bins with incomplete information. In the classical Two-Choice process a ball first queries the load of two randomly chosen bins and is then placed in the least loaded bin. In our setting,…
Maximum Likelihood Estimation of continuous variable models can be very challenging in high dimensions, due to potentially complex probability distributions. The existence of multiple interdependencies among variables can make it very…
The need to estimate a positive definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts. The coefficient and the right hand side matrices are…
The problem of estimating a random vector x from noisy linear measurements y = A x + w with unknown parameters on the distributions of x and w, which must also be learned, arises in a wide range of statistical learning and linear inverse…
We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of $m$ linear equations of the form $x_i - x_j \equiv c_{ij}\mod k$, and required to find an assignment to the $n$ variables $\{x_i\}$ that maximises…
We prove a removal lemma for systems of linear equations over finite fields: let $X_1,...,X_m$ be subsets of the finite field $\F_q$ and let $A$ be a $(k\times m)$ matrix with coefficients in $\F_q$ and rank $k$; if the linear system $Ax=b$…
A solution of linear systems of equations Ax=b and Ax=0 is a vital part of many computational packages. This paper presents a novel formulation based on the projective extension of the Euclidean space using the outer product (extended…
We present an algorithm that given a linear program with $n$ variables, $m$ constraints, and constraint matrix $A$, computes an $\epsilon$-approximate solution in $\tilde{O}(\sqrt{rank(A)}\log(1/\epsilon))$ iterations with high probability.…
Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer…