### get knitr just the way we like it

knitr::opts_chunk$set(
  message = FALSE,
  warning = FALSE,
  error = FALSE,
  tidy = FALSE,
  cache = FALSE
)

$L_1$ minimization (Linear Programming)

We solve the following problem that arises for example in sparse signal reconstruction problems such as compressed sensing: $$ \mbox{minimize } ||x||_1 \mbox{ ($L_1$) }\ \mbox{subject to } Ax = b $$

with $x\in R^n$, $A \in R^{m \times n}$ and $m\leq n.$ Reformulate the problem expressing the $L_1$ norm of $x$ as follows $$ x \leq u\ -x \leq u\ $$

where $u\in R^n$ and we minimize the sum of $u$. The reformulated problem using the stacked variables

$$ z = \begin{pmatrix}x\u\end{pmatrix} $$

is now $$ \mbox{minimize } c^{\top}z\ \mbox{subject to } \tilde{A}x = b \mbox{ (LP) }\ Gx \leq h $$ where the inequality is with respective to the positive orthant.

Here is the R code that generates a random instance of this problem and solves it.

library(ECOSolveR)
library(Matrix)
set.seed(182391)
n <- 1000L
m <- 10L
density <- 0.01
c <- c(rep(0.0, n), rep(1.0, n))

First, a function to generate random sparse matrices with normal entries.

sprandn <- function(nrow, ncol, density) {
    items <- ceiling(nrow * ncol * density)
    matrix(c(rnorm(items),
             rep(0, nrow * ncol - items)),
           nrow = nrow)
}
A <- sprandn(m, n, density)
Atilde <- Matrix(cbind(A, matrix(rep(0.0, m * n), nrow = m)), sparse = TRUE)
b <- rnorm(m)
I <- diag(n)
G <- rbind(cbind(I, -I),
           cbind(-I, -I))
G <- as(G, "dgCMatrix")
h <- rep(0.0, 2L * n)
dims <- list(l = 2L * n, q = NULL, e = 0L)

Note how ECOS expects sparse matrices, not ordinary matrices.

## Solve the problem
z <- ECOS_csolve(c = c, G = G, h = h, dims = dims, A = Atilde, b = b)

We check that the solution was found.

names(z)
z$infostring

Extract the solution.

x <- z$x[1:n]
u <- z$x[(n+1):(2*n)]
nnzx = sum(abs(x) > 1e-8)
sprintf("x reconstructed with %d non-zero entries", nnzx / length(x) * 100)


bnaras/ECOSolveR documentation built on Jan. 24, 2021, 9:37 p.m.