Tuesday, March 26, 2024

Easter 2024

 Due to Easter support and sales will be closed from Thursday, March 28th, until Monday, April 1st, both days inclusive.


The MOSEK team wishes everyone a peaceful Easter.

Thursday, March 14, 2024

Pi day and the geometric mean cone

Happy Pi Day! If Euler wasn't so keen on complex numbers then instead of $e^{i\pi}+1=0$ he would certainly have declared $$\pi^3+1=2^5$$ as the most beautiful equation. (Both sides indeed equal $32.0$). We can easily give a formal proof of this identity with MOSEK. Note that we need to simultaneously prove two inequalities $$\sqrt[3]{31}\geq \pi\quad \mathrm{and}\quad \sqrt[5]{\pi^3+1}\geq 2.$$ In MOSEK we have the geometric mean cone of dimension $n+1$: $$\mathcal{GM}^{n+1}=\{(x_1,\ldots,x_n,t)~:~x\geq 0, (x_1\cdot\ldots\cdot x_n)^{1/n}\geq |t|\},$$ so all we need to check is the feasibility of the conic problem $$(31,1,1,\pi)\in\mathcal{GM}^4,\quad (\pi^3+1,1,1,1,1,2)\in\mathcal{GM}^6.$$ This is straightforward with, for instance, Python Fusion:

M.constraint(Expr.constTerm([31, 1, 1, pi]), 

             Domain.inPGeoMeanCone())

M.constraint(Expr.constTerm([pi**3+1, 1, 1, 1, 1, 2]), 

             Domain.inPGeoMeanCone())

and voila! here is our proof:

Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 0.0000000000e+00    nrm: 1e+00    Viol.  var: 0e+00    acc: 1e-04  
  Dual.    obj: 0.0000000000e+00    nrm: 0e+00    Viol.  var: 0e+00    acc: 0e+00

Bonus question for the advanced user: which Mosek parameters do you have to modify to get this answer? Can you get away with only modifying one parameter by not too much? You can try to run it and find out yourself here.

Wednesday, March 6, 2024

Ferrari, factorizing, and portfolios in conic form

1. Suppose we want to solve the equation $$\begin{equation*}(x^2-3x+2)(x-3)(x-4)=0.\end{equation*}$$ The straightforward approach is to multiply everything out: $$\begin{equation*}x^4-10x^3+35x^2-50x+24=0\end{equation*}$$ and use the algorithm for solving a general 4-th degree equation discovered by Lodovico Ferrari, an Italian mathematician, around 1540. The procedure is pretty involved, requires solving auxiliary 3-rd degree equations and other operations. Luckily it can be summarized in explicit formulae for the quartic roots, which we don't quote here due to restricted space. Having gone through this ordeal we get, for example, that one of the four roots is $$\begin{equation*}x_1=\frac{-1}{6} \sqrt{6} \sqrt{\frac{\left(\frac{1}{2}\right)^{\frac{2}{3}} \left(\left(\frac{1}{2}\right)^{\frac{2}{3}} \left(36\sqrt{-3}+70\right)^{\frac{2}{3}}+5\left(\frac{1}{2}\right)^{\frac{1}{3}} \left(36\sqrt{-3}+70\right)^{\frac{1}{3}}+13\right)}{\left(36\sqrt{-3}+70\right)^{\frac{1}{3}}}}-\frac{1}{2} \sqrt{\frac{-1}{3} \left(\frac{1}{2}\right)^{\frac{1}{3}} \left(36\sqrt{-3}+70\right)^{\frac{1}{3}}-\frac{\frac{26}{3} \left(\frac{1}{2}\right)^{\frac{2}{3}}}{\left(36\sqrt{-3}+70\right)^{\frac{1}{3}}}+\frac{10}{3}}+\frac{5}{2}\end{equation*}$$ which numerically evaluates to $x_1\approx 1.00 - 1.38\cdot 10^{-17}i$ and with some good faith we conclude $x_1=1$. We obtain the other three roots similarly.

Of course you wouldn't do it that way. You would exploit the fact that your equation already comes almost completely factored, the small quadratic part $x^2-3x+2$ is easy to factor as $(x-1)(x-2)$ with standard school math, so we have $$\begin{equation*}(x-1)(x-2)(x-3)(x-4)=0\end{equation*}$$ and we get all the solutions for free, exactly. The only reason we used the previous method is because we thought we first have to reduce the problem to some standard form and then use it as a black-box. The black-box, however, is unnecessarily complicated for what we need here, introduces noise, and ignores the additional structure we started with so a lot of wasted work is done to go back and forth.

You can read more about the twisted story of Ferrari and his mathematical duels over polynomial equations online, for instance in this article.


2. Suppose we want to optimize a portfolio with a factor covariance model of the form $\mathrm{diag}(D)+V^TFV$, with assets $x\in \mathbb{R}^n,\ (n\approx 3000)$, where $D\in \mathbb{R}^{n}$  is specific risk, $F\in \mathbb{R}^{k\times k},\ (k\approx 40)$ is factor covariance and $V\in \mathbb{R}^{k\times n}$ is the matrix of factor exposures. The straightforward approach is to multiply everything out, in pseudocode:

Q = diag(D) + V.T @ F @ V

and use the algorithm for optimizing a general quadratic problem:

risk = sqrt(quad_form(x, Q))

The procedure is pretty involved, requires checking that the matrix $Q\in \mathbb{R}^{n\times n}$ is positive-semidefinite, which at this size is prone to accumulation of numerical errors and can be time consuming (even though we know very well $Q$ is PSD by construction). Simultaneously the modeling tool and/or conic solver will need to compute some decomposition $Q=U^TU$. Having gone through this ordeal, the solver will finally have the risk in conic form

risk = norm(U@x)

albeit working with a dense matrix $U$ with $n\times n\approx 10^7$ nonzeros will not be optimal for efficiency and numerical reasons. We finally obtain the solution.

Of course you wouldn't do it that way. You would exploit the fact that your risk model already comes almost completely factored, the small factor covariance matrix $F\in \mathbb{R}^{k\times k}$ is easy to factor using standard Cholesky decomposition as $F=H^TH$, $H\in\mathbb{R}^{k\times k}$, and we get a conic decomposition of risk for free: $$\begin{equation*}x^T(\mathrm{diag}(D)+V^TFV)x=x^T(\mathrm{diag}(D)+V^TH^THV)x=\sum_i D_ix_i^2 + \|HVx\|_2^2,\end{equation*}$$ or in pseudocode:

risk = norm([diag(D^0.5); H @ V] @ x)

The matrix $HV\in \mathbb{R}^{k\times n}$ has $kn\approx  10^5$ nonzeros (a factor of $n/k\approx 100$ fewer than before) and everything else is very sparse so we get the solution very quickly and with high accuracy. The only reason we used the previous method is because we thought we first have to formulate a general quadratic problem and then solve it as a black-box. The black-box, however, is unnecessarily complicated for what we need here, introduces noise, and ignores the additional structure we started with so a lot of wasted work is done to go back and forth.

You can read more about the efficient conic implementation of the factor model in our Portfolio Optimization Cookbook, the documentation, a CVXPY tutorial and other resources.

Monday, February 26, 2024

New Portfolio Optimization Cookbook chapter and notebook

We are happy to share that our Portfolio Optimization Cookbook has been updated with a new chapter about regression. As always the new chapter is accompanied by a notebook with an illustrative code example implemented in MOSEK Fusion for Python.

Enjoy! 

Monday, February 19, 2024

New third party interfaces to MOSEK

Recently we added PyPSA and linopy as third party interfaces to MOSEK.

PyPSA stands for Python for Power System Analysis. It is pronounced "pipes-ah. It is a toolbox for simulating and optimizing modern power systems. 

PyPSA utilizes linopy to connect to solvers, such as MOSEK. But linopy can also be used on its own. It is designed to be a link between data analysis frameworks, such as pandas and xarray, and optimization solvers, like MOSEK.  

A full list of available third party interfaces to MOSEK can be seen here. If you feel like the list is not complete, whether it is an existing interface that is missing or an interface that you wished existed, let us know! This is easiest done by contacting support@mosek.com.  

Monday, February 12, 2024

NEOS server and Mosek

MOSEK offers free licenses to academics as part of our academic initiative. This also extends to academic use of MOSEK through the NEOS server

The NEOS server is a great initiative and service hosted by Wisconsin Institute for Discovery at the University of Wisconsin in Madison. The NEOS server is a free internet-based service for solving numerical optimization problems. We at MOSEK are proud to one of the 60 supported solvers. 

Wednesday, January 24, 2024

New prices from June 2024

The new prices will come into effect on June 1st, 2024. 

The price for the basic PTS and PTON floating licenses increases with 100 USD each. Our other prices follows accordingly. With NODE licenses costing 4 times the price of their floating license counterpart and the annual maintenance 25% of the base price of the part.

This equates to a price increase of 5.1% on average.

The new prices can be found at the top of our commercial pricing page on our website.


Thursday, January 18, 2024

Seminar at LiU

 On Monday (January 22) MOSEK CEO and chief scientist Erling Dalgaard Andersen will be giving a talk at Link√∂ping University. The talk will, maybe unsurprisingly, be about conic optimization.

The seminar will take place at 13.15 in Hopningspunkten (B-huset) on the campus of LiU. It is open for everybody who are interested :)

Monday, January 15, 2024

Happy New Year!

We here at MOSEK have had a good start to the new year and we hope the same is true for all of you!

For those who have had a less than optimal start to the year we would like to share some resources that might bring you on to the fast path to optimum.

We know many users of MOSEK use it for portfolio optimization, in light of that we gladly recommend the recently published paper Markowitz Portfolio Construction at Seventy

Further there is now an unofficial package for CVX with native Apple Silicon support. This package due to the work of Dr. Wenyuan (Mike) Wang and MOSEKs own Michal Adamaszek with permission from Michael C. Grant.

We hope these resources can be of use and that you will have a great 2024!