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On the optimal objective value of random linear programs

Marzieh Bakhshi, James Ostrowski, Konstantin Tikhomirov

Abstract

We consider the problem of maximizing $\langle c,x \rangle$ subject to the constraints $Ax \leq \mathbf{1}$, where $x\in{\mathbb R}^n$, $A$ is an $m\times n$ matrix with mutually independent centered subgaussian entries of unit variance, and $c$ is a cost vector of unit Euclidean length. In the asymptotic regime $n\to\infty$, $\frac{m}{n}\to\infty$, and under some additional assumptions on $c$, we prove that the optimal objective value $z^*$ of the linear program satisfies $$ \lim\limits_{n\to\infty}\sqrt{2\log(m/n)}\,z^*= 1\quad \mbox{almost surely}. $$ In the context of high-dimensional convex geometry, our findings imply sharp asymptotic bounds on the spherical mean width of the random convex polyhedron $P=\{x\in{\mathbb R}^n:\; Ax\leq \mathbf{1}\}$. We provide numerical experiments as supporting data for the theoretical predictions. Further, we carry out numerical studies of the limiting distribution and the standard deviation of $z^*$.

On the optimal objective value of random linear programs

Abstract

We consider the problem of maximizing subject to the constraints , where , is an matrix with mutually independent centered subgaussian entries of unit variance, and is a cost vector of unit Euclidean length. In the asymptotic regime , , and under some additional assumptions on , we prove that the optimal objective value of the linear program satisfies In the context of high-dimensional convex geometry, our findings imply sharp asymptotic bounds on the spherical mean width of the random convex polyhedron . We provide numerical experiments as supporting data for the theoretical predictions. Further, we carry out numerical studies of the limiting distribution and the standard deviation of .
Paper Structure (30 sections, 23 theorems, 135 equations, 4 figures, 6 tables, 1 algorithm)

This paper contains 30 sections, 23 theorems, 135 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup and motivation
  3. Notation
  4. Theoretical guarantees
  5. Review of related literature
  6. Convex feasibility problems
  7. Random linear programs
  8. The mean width of random polyhedra
  9. Organization of the paper
  10. Preliminaries
  11. Compressible and incompressible vectors
  12. Standard concentration and anti-concentration for subgaussian variables
  13. Moderate deviations for linear combinations
  14. The lower bound on $z^*$ in subgaussian setting
  15. Auxiliary results
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $K>0$. Assume that $\lim\limits_{n\to\infty}\frac{m}{n}=\infty$, and that for each $n$, the entries of the $m\times n$ coefficient matrix $A=A(n)$ are mutually independent centered $K$--subgaussian variables of unit variances. Assume further that the non-random unit cost vectors $c=c(n)\in\mathb and, in particular, $\lim\limits_{n\to\infty} \sqrt{2\log(m/n)}\,z^* = 1$ almost surely.

Figures (4)

  • Figure 1: Structure of the proof of Theorem \ref{['th main nongauss']}
  • Figure 2: Structure of the proof of Theorem \ref{['th main gauss']}
  • Figure 3: The frequency histogram and the empirical cumulative distribution function of the optimal objective values for the Gaussian matrix $A$ with $m=1000$, $n=50$ (the sample size = $1000$). The KS test results: KS statistic = $0.0232$, $p$-value = $0.6453$.
  • Figure 4: The frequency histogram and the empirical cumulative distribution function of the optimal objective values for the Rademacher matrix $A$ with $m=1000$, $n=50$ (the sample size = $1000$). The KS test results: KS statistic = $0.0219$, $p$-value = $0.7161$.

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2: Main result: asymptotics of $z^*$ in subgaussian setting
  • Remark
  • Theorem 1.3: Asymptotics of $z^*$ in Gaussian setting
  • Corollary 1.4: The mean width in subgaussian setting
  • Corollary 1.5: The mean width in Gaussian setting
  • Definition 2.1: Sparse vectors
  • Definition 2.2: Compressible and incompressible vectors, RV2008
  • Proposition 2.3
  • Proposition 2.4
  • ...and 36 more