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Linear programming is the seminal optimization problem that has spawned and grown into today's rich and diverse optimization modeling and algorithmic landscape. This article provides an overview of the recent development of first-order…
Ahlswede and Katona (1977) posed the following isodiametric problem in Hamming spaces: For every $n$ and $1\le M\le2^{n}$, determine the minimum average Hamming distance of binary codes with length $n$ and size $M$. Fu, Wei, and Yeung…
We revisit the linear programming bounds for the size vs. distance trade-off for binary codes, focusing on the bounds for the almost-balanced case, when all pairwise distances are between $d$ and $n-d$, where $d$ is the code distance and…
Recall that a binary linear code of length $n$ is a linear subspace $\mathcal{C} = \{x\in\mathbb{F}_2^n\mid Ax=0\}$. Here the parity check matrix $A$ is a binary $m\times n$ matrix of rank $m$. We say that $\mathcal{C}$ has rate $R=1-\frac…
"The Price of Robustness" by Bertsimas and Sim represented a breakthrough in the development of a tractable robust counterpart of Linear Programming Problems. However, the central modeling assumption that the deviation band of each…
It has been shown that the parallel Lattice Linear Predicate (LLP) algorithm solves many combinatorial optimization problems such as the shortest path problem, the stable marriage problem and the market clearing price problem. In this…
The locally repairable codes (LRCs) were introduced to correct erasures efficiently in distributed storage systems. LRCs are extensively studied recently. In this paper, we first deal with the open case remained in \cite{q} and derive an…
Bilevel programming problems frequently arise in real-world applications across various fields, including transportation, economics, energy markets and healthcare. These problems have been proven to be NP-hard even in the simplest form with…
Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…
The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to…
We consider a rate-distortion problem with side information at multiple decoders. Several upper and lower bounds have been proposed for this general problem or special cases of it. We provide an upper bound for general instances of this…
We consider continuous linear programs over a continuous finite time horizon $T$, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions, where we search for optimal solutions in the space…
There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the…
In this note we develop a linear programming framework to produce upper and lower bounds for the lonely runner problem.
This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. Several methods to compute extensions of linear codes are presented. Many codes improving the…
We are concerned with linear redundancy storage schemes regarding their ability to provide concurrent (local) recovery of multiple data objects. This paper initiates a study of such systems within the classical coding theory. We show how we…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates…
An approach is established for maximizing the Lower bound on the Mismatch capacity (hereafter abbreviated as LM rate), a key performance bound in mismatched decoding, by optimizing the channel input probability distribution. Under a fixed…
In this paper, a general algorithm is proposed for rate analysis and code design of linear index coding problems. Specifically a solution for minimum rank matrix completion problem over finite fields representing the linear index coding…