## What is `CVXR`?

`CVXR` is an R package that provides an object-oriented modeling language for convex optimization, similar to `CVX`, `CVXPY`, `YALMIP`, and `Convex.jl`. It allows the user to formulate convex optimization problems in a natural mathematical syntax rather than the restrictive standard form required by most solvers. The user specifies an objective and set of constraints by combining constants, variables, and parameters using a library of functions with known mathematical properties. `CVXR` then applies signed disciplined convex programming (DCP) to verify the problem’s convexity. Once verified, the problem is converted into standard conic form using graph implementations and passed to a cone solver such as ECOS or SCS.

## Where can I learn more?

The paper by @fu:naras:boyd:2017 is the main reference. Further documentation, along with a number of tutorial examples, is also available on the CVXR website.

Below we provide a simple example to get you started.

## A Simple Example

Consider a simple linear regression problem where it is desired to estimate a set of parameters using a least squares criterion.

We generate some synthetic data where we know the model completely, that is

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

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

    set.seed(123)

    n <- 100
    p <- 10
    beta <- -4:5   # beta is just -4 through 5.

    X <- matrix(rnorm(n * p), nrow=n)
    colnames(X) <- paste0("beta_", beta)
    Y <- X %*% beta + rnorm(n)

Given the data \(X\) and \(Y\), we can estimate the \(\beta\) vector using `lm` function in R that fits a standard regression model.

    ls.model <- lm(Y ~ 0 + X)   # There is no intercept in our model above
    m <- data.frame(ls.est = coef(ls.model))
    rownames(m) <- paste0("$\\beta_{", 1:p, "}$")
    knitr::kable(m)

| ls.est
--- | ---
\(\beta_{1}\) | -3.9196886
\(\beta_{2}\) | -3.0117048
\(\beta_{3}\) | -2.1248242
\(\beta_{4}\) | -0.8666048
\(\beta_{5}\) | 0.0914658
\(\beta_{6}\) | 0.9490454
\(\beta_{7}\) | 2.0764700
\(\beta_{8}\) | 3.1272275
\(\beta_{9}\) | 3.9609565
\(\beta_{10}\) | 5.1348845

These are the least-squares estimates and can be seen to be reasonably close to the original \(\beta\) values -4 through 5.

## The `CVXR` formulation

The `CVXR` formulation states the above as an optimization problem:

\[ \begin{array}{ll} \underset{\beta}{\mbox{minimize}} & \|y - X\beta\|_2^2, \end{array} \] which directly translates into a problem that `CVXR` can solve as shown in the steps below.

- Step 0. Load the `CVXR` library

    suppressWarnings(library(CVXR, warn.conflicts=FALSE))

- Step 1. Define the variable to be estimated

    betaHat <- Variable(p)

- Step 2. Define the objective to be optimized

    objective <- Minimize(sum((Y - X %*% betaHat)^2))

Notice how the objective is specified using functions such as `sum`, `*%*` and `^`, that are familiar to R users despite that fact that `betaHat` is no ordinary R expression but a `CVXR` expression.

- Step 3. Create a problem to solve

    problem <- Problem(objective)

- Step 4. Solve it!

    result <- solve(problem)

- Step 5. Extract solution and objective value

    ## Objective value: 97.847586

We can indeed satisfy ourselves that the results we get matches that from `lm`.

    m <- cbind(coef(ls.model), result$getValue(betaHat))
    colnames(m) <- c("lm est.", "CVXR est.")
    rownames(m) <- paste0("$\\beta_{", 1:p, "}$")
    knitr::kable(m)

| lm est. | CVXR est.
--- | --- | ---
\(\beta_{1}\) | -3.9196886 | -3.9196886
\(\beta_{2}\) | -3.0117048 | -3.0117048
\(\beta_{3}\) | -2.1248242 | -2.1248242
\(\beta_{4}\) | -0.8666048 | -0.8666048
\(\beta_{5}\) | 0.0914658 | 0.0914658
\(\beta_{6}\) | 0.9490454 | 0.9490454
\(\beta_{7}\) | 2.0764700 | 2.0764700
\(\beta_{8}\) | 3.1272275 | 3.1272275
\(\beta_{9}\) | 3.9609565 | 3.9609565
\(\beta_{10}\) | 5.1348845 | 5.1348845

## Wait a minute! What have we gained?

On the surface, it appears that we have replaced one call to `lm` with at least five or six lines of new R code. On top of that, the code actually runs slower, and so it is not clear what was really achieved.

So suppose we knew for a fact that the $\beta$s were nonnegative and we wish to take this fact into account. This is nonnegative least squares regression and `lm` would no longer do the job.

In `CVXR`, the modified problem merely requires the addition of a constraint to the problem definition.

    problem <- Problem(objective, constraints = list(betaHat >= 0))
    result <- solve(problem)
    m <- data.frame(CVXR.est = result$getValue(betaHat))
    rownames(m) <- paste0("$\\beta_{", 1:p, "}$")
    knitr::kable(m)

| CVXR.est
--- | ---
\(\beta_{1}\) | 0.0000000
\(\beta_{2}\) | 0.0000000
\(\beta_{3}\) | 0.0000000
\(\beta_{4}\) | 0.0000000
\(\beta_{5}\) | 1.2374488
\(\beta_{6}\) | 0.6234665
\(\beta_{7}\) | 2.1230663
\(\beta_{8}\) | 2.8035640
\(\beta_{9}\) | 4.4448016
\(\beta_{10}\) | 5.2073521

We can verify once again that these values are comparable to those obtained from another R package, say nnls.

    if (requireNamespace("nnls", quietly = TRUE)) {
        nnls.fit <- nnls::nnls(X, Y)$x
    } else {
        nnls.fit <- rep(NA, p)
    }

    m <- cbind(result$getValue(betaHat), nnls.fit)
    colnames(m) <- c("CVXR est.", "nnls est.")
    rownames(m) <- paste0("$\\beta_{", 1:p, "}$")
    knitr::kable(m)

| CVXR est. | nnls est.
--- | --- | ---
\(\beta_{1}\) | 0.0000000 | 0.0000000
\(\beta_{2}\) | 0.0000000 | 0.0000000
\(\beta_{3}\) | 0.0000000 | 0.0000000
\(\beta_{4}\) | 0.0000000 | 0.0000000
\(\beta_{5}\) | 1.2374488 | 1.2374488
\(\beta_{6}\) | 0.6234665 | 0.6234665
\(\beta_{7}\) | 2.1230663 | 2.1230663
\(\beta_{8}\) | 2.8035640 | 2.8035640
\(\beta_{9}\) | 4.4448016 | 4.4448016
\(\beta_{10}\) | 5.2073521 | 5.2073521

## Okay that was cool, but…

As you no doubt noticed, we have done nothing that other R packages could not do.

So now suppose that we know, for some extraneous reason, that the sum of \(\beta_2\) and \(\beta_3\) is nonpositive and but all other $\beta$s are nonnegative.

It is clear that this problem would not fit into any standard package. But in `CVXR`, this is easily done by adding a few constraints.

To express the fact that \(\beta_2 + \beta_3\) is nonpositive, we construct a row matrix with zeros everywhere, except in positions 2 and 3 (for \(\beta_2\) and \(\beta_3\) respectively).

    A <- matrix(c(0, 1, 1, rep(0, 7)), nrow = 1)
    colnames(A) <- paste0("$\\beta_{", 1:p, "}$")
    knitr::kable(A)

\(\beta_{1}\) | \(\beta_{2}\) | \(\beta_{3}\) | \(\beta_{4}\) | \(\beta_{5}\) | \(\beta_{6}\) | \(\beta_{7}\) | \(\beta_{8}\) | \(\beta_{9}\) | \(\beta_{10}\)
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0

The sum constraint is nothing but \[ A\beta <= 0 \]

which we express in R as

    constraint1 <- A %*% betaHat <= 0

NOTE: The above constraint can also be expressed simply as

    constraint1 <- betaHat[2] + betaHat[3] <= 0

but it is easier working with matrices in general with `CVXR`.

For the nonnegativity for rest of the variables, we construct a \(10\times 10\) matrix \(A\) to have 1's along the diagonal everywhere except rows 2 and 3 and zeros everywhere.

    B <- diag(c(1, 0, 0, rep(1, 7)))
    colnames(B) <- rownames(B) <- paste0("$\\beta_{", 1:p, "}$")
        knitr::kable(B)

| \(\beta_{1}\) | \(\beta_{2}\) | \(\beta_{3}\) | \(\beta_{4}\) | \(\beta_{5}\) | \(\beta_{6}\) | \(\beta_{7}\) | \(\beta_{8}\) | \(\beta_{9}\) | \(\beta_{10}\)
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
\(\beta_{1}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
\(\beta_{2}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
\(\beta_{3}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
\(\beta_{4}\) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
\(\beta_{5}\) | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0
\(\beta_{6}\) | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0
\(\beta_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0
\(\beta_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0
\(\beta_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0
\(\beta_{10}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1

The constraint for positivity is \[ B\beta > 0 \]

which we express in R as

    constraint2 <- B %*% betaHat >= 0

Now we are ready to solve the problem just as before.

    problem <- Problem(objective, constraints = list(constraint1, constraint2))
    result <- solve(problem)

And we can get the estimates of \(\beta\).

    m <- data.frame(CVXR.soln = result$getValue(betaHat))
    rownames(m) <- paste0("$\\beta_{", 1:p, "}$")
    knitr::kable(m)

| CVXR.soln
--- | ---
\(\beta_{1}\) | 0.0000000
\(\beta_{2}\) | -2.8446952
\(\beta_{3}\) | -1.7109771
\(\beta_{4}\) | 0.0000000
\(\beta_{5}\) | 0.6641308
\(\beta_{6}\) | 1.1781109
\(\beta_{7}\) | 2.3286139
\(\beta_{8}\) | 2.4144893
\(\beta_{9}\) | 4.2119052
\(\beta_{10}\) | 4.9483245

This demonstrates the chief advantage of `CVXR`: flexibility. Users can quickly modify and re-solve a problem, making our package ideal for prototyping new statistical methods. Its syntax is simple and mathematically intuitive. Furthermore, `CVXR` combines seamlessly with native R code as well as several popular packages, allowing it to be incorporated easily into a larger analytical framework. The user is free to construct statistical estimators that are solutions to a convex optimization problem where there may not be a closed form solution or even an implementation. Such solutions can then be combined with resampling techniques like the bootstrap to estimate variability.

## References