## Generate a random non-trivial linear program
set.seed(1)
m <- 15
n <- 10
s0 <- rnorm(m)
lamb0 <- pmax(-s0, 0)
s0 <- pmax(s0, 0)
x0 <- rnorm(n)
A <- matrix(rnorm(m * n), nrow = m, ncol = n)
b <- A %*% x0 + s0
c_vec <- -t(A) %*% lamb0
## Define and solve the CVXR problem
x <- Variable(n)
constraints <- list(A %*% x <= b)
prob <- Problem(Minimize(t(c_vec) %*% x), constraints)
result <- psolve(prob)
check_solver_status(prob)Linear Program
Introduction
A linear program is an optimization problem with a linear objective and affine inequality constraints. A common standard form is the following:
\[ \begin{array}{ll} \mbox{minimize} & c^Tx \\ \mbox{subject to} & Ax \leq b. \end{array} \]
Here \(A \in \mathcal{R}^{m \times n}\), \(b \in \mathcal{R}^m\), and \(c \in \mathcal{R}^n\) are problem data and \(x \in \mathcal{R}^{n}\) is the optimization variable. The inequality constraint \(Ax \leq b\) is elementwise.
For example, we might have \(n\) different products, each constructed out of \(m\) components. Each entry \(A_{ij}\) is the amount of component \(i\) required to build one unit of product \(j\). Each entry \(b_i\) is the total amount of component \(i\) available. We lose \(c_j\) for each unit of product \(j\) (\(c_j < 0\) indicates profit). Our goal then is to choose how many units of each product \(j\) to make, \(x_j\), in order to minimize loss without exceeding our budget for any component.
In addition to a solution \(x^\star\), we obtain a dual solution \(\lambda^\star\). A positive entry \(\lambda^\star_i\) indicates that the constraint \(a_i^Tx \leq b_i\) holds with equality for \(x^\star\) and suggests that changing \(b_i\) would change the optimal value.
Example
In the following code, we solve a linear program with CVXR.
## Print result
cat(sprintf("The optimal value is %f\n", result))
cat("A solution x is\n")
print(value(x))
cat("A dual solution is\n")
print(dual_value(constraints[[1]]))The optimal value is -7.834460
A solution x is
[,1]
[1,] 0.30516488
[2,] 0.36503940
[3,] 0.87203536
[4,] 0.84588014
[5,] 0.68740424
[6,] 0.78488970
[7,] 0.85640199
[8,] 0.01747114
[9,] -2.10062721
[10,] 0.73176290
A dual solution is
[1] 6.264538e-01 3.200369e-10 8.356286e-01 5.319856e-11 9.657552e-10
[6] 8.204684e-01 5.749221e-10 3.474298e-10 2.630130e-10 3.053884e-01
[11] 6.055836e-11 1.464662e-10 6.212406e-01 2.214700e+00 2.417571e-10Session Info
R version 4.5.3 (2026-03-11)
Platform: aarch64-apple-darwin20
Running under: macOS Tahoe 26.3.1
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.1
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Los_Angeles
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] CVXR_1.8.1
loaded via a namespace (and not attached):
[1] slam_0.1-55 cli_3.6.5 knitr_1.51 ECOSolveR_0.6.1
[5] rlang_1.1.7 xfun_0.56 clarabel_0.11.2 otel_0.2.0
[9] gurobi_13.0-1 Rglpk_0.6-5.1 highs_1.12.0-3 cccp_0.3-3
[13] scs_3.2.7 S7_0.2.1 jsonlite_2.0.0 backports_1.5.0
[17] rprojroot_2.1.1 htmltools_0.5.9 Rmosek_11.1.1 gmp_0.7-5.1
[21] piqp_0.6.2 rmarkdown_2.30 grid_4.5.3 evaluate_1.0.5
[25] fastmap_1.2.0 yaml_2.3.12 compiler_4.5.3 codetools_0.2-20
[29] htmlwidgets_1.6.4 Rcpp_1.1.1 here_1.0.2 osqp_1.0.0
[33] lattice_0.22-9 digest_0.6.39 checkmate_2.3.4 Matrix_1.7-4
[37] tools_4.5.3