Appendix A — Using a Commercial Solver: MOSEK

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.

A.1 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 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.

A.2 Getting MOSEK

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

Get a license. Request a free personal academic license (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.

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

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.

Check it works:
```
library(Rmosek)
mosek_version()
```

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

A.3 Using MOSEK in CVXR

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

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.4 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:

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

[1]  1.768  0.907  0.562 -0.851 -1.705

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:

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)

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 (Chapter 8):

────────────────────────────── 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.

A.5 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.