Introduction to CVXR

A Simple Example: Least Squares

Consider a simple linear regression problem where we want to estimate parameters using a least squares criterion.

We generate synthetic data where we know the true model:

\[ Y = X\beta + \epsilon \]

where \(Y\) is a \(100 \times 1\) vector, \(X\) is a \(100 \times 10\) matrix, \(\beta = (-4, -3, \ldots, 5)^\top\) is a \(10 \times 1\) vector, and \(\epsilon \sim N(0, 1)\).

set.seed(123)
n <- 100
p <- 10
beta <- -4:5

X <- matrix(rnorm(n * p), nrow = n)
Y <- X %*% beta + rnorm(n)

Using base R, we can estimate \(\beta\) via lm:

ls.model <- lm(Y ~ 0 + X)

The CVXR formulation

The same problem can be expressed as:

\[ \underset{\beta}{\text{minimize}} \quad \|Y - X\beta\|_2^2 \]

In CVXR, this translates directly:

library(CVXR)
#> 
#> Attaching package: 'CVXR'
#> The following objects are masked from 'package:stats':
#> 
#>     power, sd, var
#> The following objects are masked from 'package:base':
#> 
#>     norm, outer

betaHat <- Variable(p)
objective <- Minimize(sum((Y - X %*% betaHat)^2))
problem <- Problem(objective)
result <- psolve(problem, solver = "CLARABEL")

The optimal value and estimated coefficients:

cat("Optimal value:", result, "\n")
#> Optimal value: 97.84759
cbind(CVXR = round(value(betaHat), 3),
      lm   = round(coef(ls.model), 3))
#>                lm
#> X1  -3.920 -3.920
#> X2  -3.012 -3.012
#> X3  -2.125 -2.125
#> X4  -0.867 -0.867
#> X5   0.091  0.091
#> X6   0.949  0.949
#> X7   2.076  2.076
#> X8   3.127  3.127
#> X9   3.961  3.961
#> X10  5.135  5.135

Adding Constraints

The real power of CVXR is the ability to add constraints easily.

Nonnegative Least Squares

Suppose we know the \(\beta\)s should be nonnegative:

problem <- Problem(objective, constraints = list(betaHat >= 0))
result <- psolve(problem, solver = "CLARABEL")
round(value(betaHat), 3)
#>        [,1]
#>  [1,] 0.000
#>  [2,] 0.000
#>  [3,] 0.000
#>  [4,] 0.000
#>  [5,] 1.237
#>  [6,] 0.623
#>  [7,] 2.123
#>  [8,] 2.804
#>  [9,] 4.445
#> [10,] 5.207

Custom Constraints

Now suppose \(\beta_2 + \beta_3 \le 0\) and all other \(\beta\)s are nonnegative:

A <- matrix(c(0, 1, 1, rep(0, 7)), nrow = 1)
B <- diag(c(1, 0, 0, rep(1, 7)))

constraint1 <- A %*% betaHat <= 0
constraint2 <- B %*% betaHat >= 0

problem <- Problem(objective, constraints = list(constraint1, constraint2))
result <- psolve(problem, solver = "CLARABEL", verbose = TRUE) ## verbose = TRUE for details
#> ────────────────────────────────── CVXR v1.8.1 ─────────────────────────────────
#> ℹ Problem: 1 variable, 2 constraints (QP)
#> ℹ Compilation: "CLARABEL" via CVXR::Dcp2Cone -> CVXR::CvxAttr2Constr -> CVXR::ConeMatrixStuffing -> CVXR::Clarabel_Solver
#> ℹ Compile time: 0.013s
#> ─────────────────────────────── Numerical solver ───────────────────────────────
#> ──────────────────────────────────── Summary ───────────────────────────────────
#> ✔ Status: optimal
#> ✔ Optimal value: 1287.63
#> ℹ Compile time: 0.013s
#> ℹ Solver time: 0.001s
round(value(betaHat), 3)
#>         [,1]
#>  [1,]  0.000
#>  [2,] -2.845
#>  [3,] -1.711
#>  [4,]  0.000
#>  [5,]  0.664
#>  [6,]  1.178
#>  [7,]  2.329
#>  [8,]  2.414
#>  [9,]  4.212
#> [10,]  4.948

This demonstrates the chief advantage of CVXR: flexibility. Users can quickly modify and re-solve a problem, making the package ideal for prototyping new statistical methods. Its syntax is simple and mathematically intuitive.

Available Solvers

CVXR supports multiple solvers:

installed_solvers()
#>  [1] "CLARABEL" "SCS"      "OSQP"     "HIGHS"    "MOSEK"    "GUROBI"  
#>  [7] "GLPK"     "GLPK_MI"  "ECOS"     "ECOS_BB"  "CPLEX"    "CVXOPT"  
#> [13] "PIQP"

You can specify a solver explicitly:

psolve(problem, solver = "CLARABEL")

Session Info

sessionInfo()
#> R version 4.5.2 (2025-10-31)
#> 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  gurobi_13.0-1   
#>  [9] otel_0.2.0       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   Rcplex_0.3-8    
#> [17] backports_1.5.0  Rmosek_11.1.1    htmltools_0.5.9  sass_0.4.10     
#> [21] gmp_0.7-5.1      piqp_0.6.2       rmarkdown_2.30   grid_4.5.2      
#> [25] evaluate_1.0.5   jquerylib_0.1.4  fastmap_1.2.0    yaml_2.3.12     
#> [29] lifecycle_1.0.5  compiler_4.5.2   codetools_0.2-20 Rcpp_1.1.1      
#> [33] osqp_1.0.0       lattice_0.22-9   digest_0.6.39    R6_2.6.1        
#> [37] bslib_0.10.0     checkmate_2.3.4  Matrix_1.7-4     tools_4.5.2     
#> [41] cachem_1.1.0

Introduction to CVXR

Anqi Fu and Balasubramanian Narasimhan

2026-03-05

Overview