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)## [1] "x" "y" "s" "z" "infostring"
## [6] "retcodes" "summary" "timing"
z$infostring## [1] "Optimal solution found"Extract the solution.
x <- z$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)## [1] "x reconstructed with 1 non-zero entries"