# Introduction to CVXR

#### Anqi Fu and Balasubramanian Narasimhan

#### 2026-08-23

## Overview

`CVXR` is an R package that provides an object-oriented modeling language for convex optimization, similar to CVXPY for Python. It allows you to formulate and solve convex optimization problems in a natural mathematical syntax.

## 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':
    #>
    #>     convolve, power, sd, var
    #> The following objects are masked from 'package:base':
    #>
    #>     diag, norm, outer

    betaHat <- Variable(p)
    objective <- Minimize(sum((Y - X %*% betaHat)^2))
    problem <- Problem(objective)
    result <- psolve(problem) ## use default solver

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 v1.9.2 ─────────────────────────────────
    #> 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: 0.013s
    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 15 solvers, both open source and commercial: Clarabel, SCS, OSQP, HiGHS, MOSEK, Gurobi, GLPK, GLPK_MI, ECOS, ECOS_BB, CPLEX, CVXOPT, PIQP, SCIP, and XPRESS. Smooth nonlinear programs additionally use the IPOPT and UNO solvers.

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

You can specify a solver explicitly:

    psolve(problem, solver = "CLARABEL")

## What’s New

Recent releases add disciplined nonlinear programming (`psolve(prob, nlp = TRUE)`), bounds propagation through expressions (`get_bounds()`), a derivative / sensitivity-analysis API (`requires_grad = TRUE`), and new atoms such as `convolve()`. CVXR also supports element-wise matrix indexing using R’s native idioms:

    ind <- which(!is.na(Rmiss), arr.ind = TRUE)
    prob <- Problem(Minimize(obj), list(X[ind] == Rmiss[ind]))

For a release-by-release summary see `vignette("whats_new")`, and `news(package = "CVXR")` for the full details.

## Further Reading

- The CVXR website has many worked examples
- The CVXPY documentation covers the underlying mathematical framework
- The published paper: Fu, Narasimhan, and Boyd (2020). “CVXR: An R Package for Disciplined Convex Optimization.” Journal of Statistical Software, 94(14), DOI:10.18637/jss.v094.i14.

## Session Info

    sessionInfo()
    #> R version 4.6.1 (2026-06-24)
    #> Platform: aarch64-apple-darwin23
    #> Running under: macOS Tahoe 26.6.2
    #>
    #> Matrix products: default
    #> BLAS:   /Library/Frameworks/R.framework/Versions/4.6/Resources/lib/libRblas.0.dylib
    #> LAPACK: /Library/Frameworks/R.framework/Versions/4.6/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_1.9.2
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] Matrix_1.7-6     piqp_0.6.2       jsonlite_2.0.0   compiler_4.6.1
    #>  [5] highs_1.14.0-2   Rcpp_1.1.2       slam_0.1-56      cccp_0.3-3
    #>  [9] jquerylib_0.1.4  yaml_2.3.12      fastmap_1.2.0    clarabel_0.11.3
    #> [13] lattice_0.22-9   R6_2.6.1         scip_1.10.0-3    knitr_1.51
    #> [17] backports_1.5.1  Rcplex_0.3-8     checkmate_2.3.4  gurobi_13.0-1
    #> [21] osqp_1.0.0       bslib_0.12.0     rlang_1.3.0      cachem_1.1.0
    #> [25] xfun_0.60        sass_0.4.10      S7_0.2.2         otel_0.2.0
    #> [29] cli_3.6.6        Rglpk_0.6-5.1    digest_0.6.39    grid_4.6.1
    #> [33] xpress_9.8.2     gmp_0.7-5.1      lifecycle_1.0.5  ECOSolveR_0.6.1
    #> [37] scs_3.2.7        evaluate_1.0.5   codetools_0.2-20 Rmosek_11.1.2
    #> [41] rmarkdown_2.31   tools_4.6.1      htmltools_0.5.9