Semidefinite Program

Introduction

A semidefinite program (SDP) is an optimization problem of the form

$$\begin{array}{ll}\text{minimize}& \mathbf{tr}(CX)\\ \text{subject to}& \mathbf{tr}(A_{i}X)=b_i,i=1,\dots ,p\\ & X⪰0,\end{array}$$

where $\mathbf{tr}$ is the trace function, $X\in {\mathcal{S}}^n$ is the optimization variable and $C,A_1,\dots ,A_p\in {\mathcal{S}}^n$, and $b_1,\dots ,b_p\in \mathcal{R}$ are problem data, and $X⪰0$ is a matrix inequality. Here ${\mathcal{S}}^n$ denotes the set of $n$-by-$n$ symmetric matrices.

An example of an SDP is to complete a covariance matrix $\tilde{\mathrm{\Sigma }}\in {\mathcal{S}}_{+}^n$ with missing entries $M\subset \{1,\dots ,n\}\times \{1,\dots ,n\}$:

$$\begin{array}{ll}\text{minimize}& 0\\ \text{subject to}& {\mathrm{\Sigma }}_{ij}={\tilde{\mathrm{\Sigma }}}_{ij},(i,j)\notin M\\ & \mathrm{\Sigma }⪰0,\end{array}$$

Example

In the following code, we solve a SDP with CVXR.

## Problem dimensions
set.seed(4)
n <- 3   # matrix size
p <- 3   # number of equality constraints

## Generate random problem data
C <- matrix(rnorm(n^2), nrow = n)
sdp_data <- lapply(1:p, function(i) {
    list(A = matrix(rnorm(n^2), nrow = n), b = rnorm(1))
})

## Define and solve the CVXR problem
X <- Variable(c(n, n), symmetric = TRUE)
## Trace constraints: tr(A_i X) = b_i
trace_constraints <- lapply(sdp_data, function(s) {
    matrix_trace(s$A %*% X) == s$b
})
prob <- Problem(Minimize(matrix_trace(C %*% X)),
                constraints = c(list(PSD(X)), trace_constraints))
result <- psolve(prob)
check_solver_status(prob)

## Print result
cat(sprintf("The optimal value is %f\n", result))
cat("A solution X is\n")
print(value(X))

The optimal value is 1.395478
A solution X is
           [,1]       [,2]        [,3]
[1,]  0.3771715 -0.5692174  0.11666372
[2,] -0.5692174  0.8590481 -0.17606590
[3,]  0.1166637 -0.1760659  0.03608551

Session Info

R version 4.5.2 (2025-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS Tahoe 26.3

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 datasets  utils     methods   base     

other attached packages:
[1] CVXR_1.8.0.9207

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] Rcplex_0.3-8      Rmosek_11.1.1     rprojroot_2.1.1   htmltools_0.5.9  
[21] gmp_0.7-5.1       piqp_0.6.2        rmarkdown_2.30    grid_4.5.2       
[25] evaluate_1.0.5    fastmap_1.2.0     yaml_2.3.12       compiler_4.5.2   
[29] codetools_0.2-20  htmlwidgets_1.6.4 Rcpp_1.1.1        here_1.0.2       
[33] osqp_1.0.0        lattice_0.22-9    digest_0.6.39     checkmate_2.3.4  
[37] Matrix_1.7-4      tools_4.5.2      

References