### get knitr just the way we like it knitr::opts_chunk$set( message = FALSE, warning = FALSE, error = FALSE, tidy = FALSE, cache = FALSE )
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)
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