appendix-mosek.qmd

# Using a Commercial Solver: MOSEK {#sec-mosek}

<!-- STATIC solver demo. CVXR's open-source solvers handle everything in this book;
     this short chapter shows how to plug in MOSEK (free for academics) for the cases
     where a commercial conic solver helps. The solve chunk is `eval: false` on purpose
     so the chapter renders without MOSEK installed. The captured output is genuine:
     a 5x5 nearest-PSD SDP solved with MOSEK 11.1.2 on 2026-06-10. The CVXR summary is
     real psolve(verbose=TRUE) output; MOSEK's native log is real Rmosek verbose=10
     output (CVXR will stream it inline under verbose in a future release --- see
     new_design/features/TODO.md). -->

::: callout-note
## What you'll be able to do
Install and license **MOSEK** --- a commercial-grade conic solver that is **free for
academic use** --- and route any CVXR problem to it by changing a single word. You
will also see what MOSEK's detailed solver log looks like on a small semidefinite
program.
:::

```{r}
#| label: setup
#| include: false
library(CVXR)
```

## Why a commercial solver?

Every example in this book is solved with the open-source solvers that ship with
CVXR --- Clarabel, OSQP, SCS, HiGHS --- and for the overwhelming majority of problems
they are excellent. But commercial solvers can pull ahead on large or numerically hard
problems, especially **semidefinite** and **second-order-cone** programs, and they
tend to report richer diagnostics. [MOSEK](https://www.mosek.com) is a standout: a
mature interior-point solver that is **free for students and academic researchers**.

The good news is that *switching solvers changes nothing about your model.* You write
the problem exactly as before; you only name a different solver at solve time.

## Getting MOSEK

MOSEK is commercial software, but academic access is free and the setup is quick:

1. **Get a license.** Request a free
   [personal academic license](https://www.mosek.com/products/academic-licenses/)
   (or a 30-day commercial trial). Save the license file to `~/mosek/mosek.lic`
   --- the default location MOSEK searches --- or point the `MOSEKLM_LICENSE_FILE`
   environment variable at it.

2. **Install the `Rmosek` interface.** `Rmosek` is distributed by MOSEK, not from
   CRAN, and is built against your local MOSEK installation:

   ```r
   install.packages("Rmosek", repos = "https://download.mosek.com/stable/rmosek/")
   library(Rmosek)
   mosek_attachbuilder()   # locate the MOSEK Optimization Suite
   install.rmosek()        # build and install the interface
   ```

   The official, platform-by-platform guide is at
   [docs.mosek.com](https://docs.mosek.com/latest/rmosek/install-interface.html).

3. **Check it works:**

   ```r
   library(Rmosek)
   mosek_version()
   ```

Once `Rmosek` loads and the license is found, CVXR will discover MOSEK automatically.

## Using MOSEK in CVXR

Selecting MOSEK is a one-word change to `psolve()`:

```r
psolve(prob, solver = "MOSEK")
```

That is the whole API. Variables, constraints, the objective, and dual-variable
recovery are all identical to every other solver --- the solver name is the only thing
that changes.

## A worked example: the nearest PSD matrix

Here is a small **semidefinite program**, the kind of problem where a commercial conic
solver earns its keep. Given a symmetric (but indefinite) matrix $M$, we look for the
positive-semidefinite matrix $X$ closest to it in Frobenius norm:

$$
\underset{X \succeq 0}{\text{minimize}}\ \ \|X - M\|_F^2 .
$$

The constraint $X \succeq 0$ ("$X$ is PSD") is declared right on the variable:

```{r}
#| label: data
set.seed(1)
n <- 5
M <- matrix(rnorm(n * n), n, n)
M <- (M + t(M)) / 2          # a symmetric --- but indefinite --- target matrix
round(eigen(M, only.values = TRUE)$values, 3)   # note the negative eigenvalues
```

The target has negative eigenvalues, so it is not itself PSD. We solve for the closest
matrix that is, and turn on `verbose = TRUE` to watch MOSEK work:

```{r}
#| label: mosek-solve
#| eval: false
X    <- Variable(c(n, n), PSD = TRUE)            # symmetric PSD matrix variable
prob <- Problem(Minimize(sum_squares(X - M)))    # nearest PSD matrix, Frobenius norm
psolve(prob, solver = "MOSEK", verbose = TRUE)
```

::: callout-tip
## This chunk is pre-rendered
The solve above is marked `eval: false` so the book builds without MOSEK installed.
The output below is the genuine result of running it with MOSEK 11.1.2.
:::

With `verbose = TRUE`, CVXR reports the compilation and the solve summary --- note that
the chosen solver is `MOSEK`, and that compile time and solver time are tracked
separately (@sec-dpp):

```
────────────────────────────── CVXR v1.9.1 ──────────────────────────────
ℹ Problem: 1 variable, 0 constraints (DCP)
ℹ Compilation: "MOSEK" via CVXR::Dcp2Cone -> CVXR::CvxAttr2Constr ->
               CVXR::ConeMatrixStuffing -> CVXR::Mosek_Solver
ℹ Compile time: 0.21s
─────────────────────────── Numerical solver ────────────────────────────
──────────────────────────────── Summary ────────────────────────────────
✔ Status: optimal
✔ Optimal value: 3.6305
ℹ Compile time: 0.21s
ℹ Solver time: 0.019s
```

MOSEK also keeps its **own** detailed log of the solve --- the problem type, the
interior-point iteration table, and a final optimizer summary. A future CVXR release
will stream this log inline whenever you pass `verbose = TRUE` (as CVXPY does); here is
what MOSEK reports for this SDP:

```
Problem
  Name                   :
  Objective sense        : minimize
  Type                   : CONIC (conic optimization problem)
  Constraints            : 0
  Affine conic cons.     : 2 (42 rows)
  Scalar variables       : 16
  Matrix variables       : 1 (scalarized: 1)
  Integer variables      : 0

Optimizer  - threads                : 16
Optimizer  - solved problem         : the primal
Optimizer  - Constraints            : 26
Optimizer  - Cones                  : 1
Optimizer  - Scalar variables       : 28                conic       : 27
Optimizer  - Semi-definite variables: 1                 scalarized  : 15
Factor     - nonzeros before factor : 351               after factor: 351
Factor     - dense dim.             : 0                 flops       : 7.88e+03
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME
0   1.6e+00  1.0e+00  2.0e+00  0.00e+00   0.000000000e+00   -1.000000000e+00  1.0e+00  0.00
1   4.1e-01  2.5e-01  5.8e-01  -3.80e-01  1.089704904e+00   1.312151618e+00   2.5e-01  0.00
2   5.9e-02  3.6e-02  1.1e-02  5.00e-01   3.386613408e+00   3.242645938e+00   3.6e-02  0.00
3   7.4e-03  4.6e-03  5.1e-04  9.41e-01   3.608929356e+00   3.590619106e+00   4.6e-03  0.00
4   1.0e-03  6.4e-04  3.3e-05  1.01e+00   3.627235469e+00   3.624865947e+00   6.4e-04  0.00
5   3.2e-05  2.0e-05  2.3e-07  1.00e+00   3.630317873e+00   3.630261513e+00   2.0e-05  0.00
6   4.0e-06  2.5e-06  1.1e-08  1.00e+00   3.630469253e+00   3.630462235e+00   2.5e-06  0.00
7   4.0e-07  2.5e-07  3.7e-10  1.00e+00   3.630493170e+00   3.630492590e+00   2.5e-07  0.00
8   1.2e-08  7.5e-09  2.0e-12  1.00e+00   3.630495992e+00   3.630495975e+00   7.5e-09  0.00
Optimizer terminated. Time: 0.00

Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 3.6304959921e+00    nrm: 4e+00    Viol.  var: 0e+00   barvar: 0e+00
  Dual.    obj: 3.6304959749e+00    nrm: 2e+00    Viol.  var: 2e-11   barvar: 1e-24
Optimizer summary
  Optimizer              -                    time: 0.00
    Interior-point       - iterations : 8     time: 0.00
    Simplex              - iterations : 0     time: 0.00
    Mixed integer        - relaxations: 0     time: 0.00
```

Read the iteration table top to bottom: MOSEK drives the primal/dual feasibility
residuals (`PFEAS`, `DFEAS`) and the duality gap (`MU`) down toward zero, the primal and
dual objectives (`POBJ`, `DOBJ`) squeeze together, and after eight interior-point
iterations they meet at the optimal value **3.6305** --- exactly the number in CVXR's
summary. The solution $X^\star$ is the original matrix with its negative eigenvalues
clipped to zero, which is the classic closed form for this problem; CVXR and MOSEK
recover it numerically without you having to know that.

## Takeaways

- CVXR's bundled open-source solvers cover everything in this book; reach for a
  **commercial solver** mainly for large or hard **conic / semidefinite** problems.
- **MOSEK is free for academics.** Get a license, install `Rmosek` from MOSEK's repo,
  and CVXR finds it automatically.
- Switching solvers is a **one-word change** --- `psolve(prob, solver = "MOSEK")` ---
  with no change to the model.
- `verbose = TRUE` reports CVXR's compile/solve summary; MOSEK additionally produces a
  detailed interior-point log of its own.

For the full list of solvers CVXR can drive --- open-source and commercial --- and
their capabilities, see the [CVXR solver documentation](https://cvxr.rbind.io).