Semidefinite o-pi-timality

Happy $\pi$ Day! If other methods fail, you can always compute $\pi$ with the MOSEK semidefinite optimizer. We leave the details as an interesting exercise for the curious readers. Some hints are hidden in our Modeling Cookbook .

	from mosek.fusion import *
	import mosek.fusion.pythonic
	import numpy as np
	def showmepi(d, m):
	A = np.eye(m, k = 1) - np.eye(m)
	with Model() as M:
	S = M.variable(Domain.inPSDCone(m))
	M.constraint(Expr.dot(S, np.eye(m)) == 1)
	M.objective(ObjectiveSense.Maximize, Expr.dot(S, A + A.T))
	M.solve()
	print(f'{np.sqrt(-M.primalObjValue())*(m+1):.{d}f}')
	for d, m in enumerate([5, 20, 200, 400, 600, 1600]):
	showmepi(d + 1, m)
	'''
	output:
	3.1
	3.14
	3.142
	3.1416
	3.14159
	3.141592
	'''

view raw showmepi.py hosted with ❤ by GitHub

Using MOSEK with CVX

Due to popular demand we present the full modern installation process of CVX+MOSEK. It works the same way on all platforms supported by MOSEK.

If you experience issues with CVX+MOSEK please reinstall from scratch following these instructions. If you already did that, and there are still issues then please contact us with your platform, MOSEK version, license type, and an explanation of which step failed including full log/error messages.

MOSEK support is unable to help with old, broken, manually altered and other CVX installations that didn't follow this process. In particular please don't use the older 2020 CVX version which comes with included, now quite outdated, MOSEK 9.1. 

Step 1. Installing CVX

1. Download and unpack the open-source CVX 2.2 release from the first paragraph ("Effective April 23, 2024") of https://cvxr.com/cvx/download/ . Ignore the legacy download matix further down. An explicit download link for the latest release as of March 2025 is https://github.com/cvxr/CVX/releases/tag/2.2.2
2. Navigate to the unpacked installation in MATLAB and run "cvx_setup", as explained in https://cvxr.com/cvx/doc/install.html
3. The log output should indicate success and the free solvers like SeDuMi and SDPT3 should be detected.

Step 2. Installing MOSEK

1. Download and install MOSEK for your platform following https://docs.mosek.com/latest/install/installation.html#general-setup Make sure to perform all the steps, for instance on OSX running a python installation script is needed, and on Windows for a manual installation (unpacking a ZIP file) manually setting the environment variable PATH is needed.
2. Obtain a MOSEK license and install it according to the instructions in the email or https://docs.mosek.com/latest/licensing/quickstart.html#i-have-a-license-file For most users of personal academic and trial licenses the default location will be sufficient. If you have a different license type (for example floating) configure it according to the manual.
3. (Optionally) run the "msktestlic" script in the bin folder of the MOSEK installation to test that license is set up correctly. This is not a MATLAB command, but a script to run in the terminal/command line.
4. Using "addpath" in MATLAB add the MOSEK toolbox to the MATLAB path, as shown in https://docs.mosek.com/latest/toolbox/install-interface.html
5. In MATLAB run the "mosekdiag" command to verify that MOSEK works in MATLAB as in https://docs.mosek.com/latest/toolbox/install-interface.html#testing-the-installation . In case of errors read the messages carefully and fix the errors. See https://docs.mosek.com/latest/toolbox/install-interface.html#troubleshooting for additional explanations for typical issues.

Step 3. Configuring MOSEK in CVX.

At this point you have verified that both CVX and MOSEK work in MATLAB and all that is left is to combine them together.

Making sure that MOSEK is still in your MATLAB path navigate to the CVX installation folder and run "cvx_setup". In the log you should see that MOSEK is detected and configured, in addition to the free solvers.

Warning. NEVER use ''cvx_precision", and especially "cvx_precision best" with MOSEK. It won't do any good and in the worst case will lead to nonsense results. If you really need to change solver termination tolerances do it by setting explicit MOSEK parameters, but first read "Should MOSEK parameters be tweaked?" on our blog.