`vignettes/cvxr_intro.Rmd`

`cvxr_intro.Rmd`

`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.

The paper by Fu, Narasimhan, and 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.

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.

`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

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 |

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 |

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.

Fu, Anqi, Balasubramanian Narasimhan, and Stephen Boyd. 2017.
“CVXR: An r Package for Disciplined Convex Optimization.”
https://web.stanford.edu/~boyd/papers/cvxr_paper.html.