## Generate a random non-trivial quadratic program.
## Construct h and b from a known feasible point x0
## to guarantee joint feasibility of the constraints.
set.seed(1)
m <- 15
n <- 10
p <- 5
P <- matrix(rnorm(n * n), nrow = n, ncol = n)
P <- t(P) %*% P
q <- rnorm(n)
G <- matrix(rnorm(m * n), nrow = m, ncol = n)
A <- matrix(rnorm(p * n), nrow = p, ncol = n)
x0 <- rnorm(n)
h <- G %*% x0 + abs(rnorm(m))
b <- A %*% x0
## Define and solve the CVXR problem
x <- Variable(n)
ineq_con <- G %*% x <= h
eq_con <- A %*% x == b
prob <- Problem(Minimize(0.5 * quad_form(x, P) + t(q) %*% x),
constraints = list(ineq_con, eq_con))
result <- psolve(prob)
check_solver_status(prob)Quadratic Program
Introduction
A quadratic program is an optimization problem with a quadratic objective and affine equality and inequality constraints. A common standard form is the following:
\[ \begin{array}{ll} \mbox{minimize} & (1/2)x^TPx + q^Tx\\ \mbox{subject to} & Gx \leq h \\ & Ax = b. \end{array} \]
Here \(P \in \mathcal{S}^{n}_+\), \(q \in \mathcal{R}^n\), \(G \in \mathcal{R}^{m \times n}\), \(h \in \mathcal{R}^m\), \(A \in \mathcal{R}^{p \times n}\), and \(b \in \mathcal{R}^p\) are problem data and \(x \in \mathcal{R}^{n}\) is the optimization variable. The inequality constraint \(Gx \leq h\) is elementwise.
A simple example of a quadratic program arises in finance. Suppose we have \(n\) different stocks, an estimate \(r \in \mathcal{R}^n\) of the expected return on each stock, and an estimate \(\Sigma \in \mathcal{S}^{n}_+\) of the covariance of the returns. Then we solve the optimization problem
\[ \begin{array}{ll} \mbox{minimize} & (1/2)x^T\Sigma x - r^Tx\\ \mbox{subject to} & x \geq 0 \\ & \mathbf{1}^Tx = 1, \end{array} \]
to find a portfolio allocation \(x \in \mathcal{R}^n_+\) that optimally balances expected return and variance of return.
When we solve a quadratic program, in addition to a solution \(x^\star\), we obtain a dual solution \(\lambda^\star\) corresponding to the inequality constraints. A positive entry \(\lambda^\star_i\) indicates that the constraint \(g_i^Tx \leq h_i\) holds with equality for \(x^\star\) and suggests that changing \(h_i\) would change the optimal value.
Example
In the following code, we solve a quadratic 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 corresponding to the inequality constraints is\n")
print(dual_value(ineq_con))The optimal value is 10.511135
A solution x is
[,1]
[1,] 1.02038881
[2,] 0.65071749
[3,] -0.21167231
[4,] -0.54479000
[5,] 0.00227552
[6,] 1.03862768
[7,] -0.20833046
[8,] 0.44450188
[9,] -1.15910026
[10,] 0.27972523
A dual solution corresponding to the inequality constraints is
[1] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.303641 2.879660
[9] 0.000000 0.000000 2.467416 0.000000 0.000000 0.000000 8.135815Session 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