Thursday, December 13, 2018

Christmas tree denoising via SDP

In our last year's Christmas special we ran logistic regression on a sampled Christmas tree. By public demand this year we treat the tree with some semidefinite programming.

We start with a $50 \times 100$ binary image with noise.

It is convenient to encode the image as a sequence of pixels $x_i\in\{-1,1\}$. Then we can write (see eg. here) a simple binary quadratic denoising model
$$\mathrm{maximize} \sum_{i\sim j}z_iz_j+\gamma\sum_i x_iz_i,\quad z_i\in\{-1,1\}$$
which favors similarity between neighboring ($i\sim j$) pixels in $z$ and the similarity between $z$ and the original $x$, with tradeoff between the two measures provided by $\gamma$.

Binary quadratic problems have a natural SDP relaxation, in this case 
$$ \begin{array}{ll} \mathrm{mazimize} & \sum_{i\sim j} Z_{ij} +\gamma x^Tz \\ \mathrm{s.t.} & \left[ \begin{array}{cc}Z&z\\z^T&1\end{array} \right] \succeq 0, \\ & \mathrm{diag}(Z)=1.\end{array} $$
We solve the SDP for various $\gamma$ ($Z$ has dimension $5000$; this is not the most efficient image processing algorithm ever invented). To further simplify we don't round but let the raw $z$ interpolate between shades of green. 

Merry Christmas from the MOSEK team.

Tuesday, November 27, 2018

Geometric programming preview

Geometric programs (GP) can be conveniently expressed in conic form using the exponential cone
$$ x\geq y\exp(z/y),\quad y>0$$
in the upcoming version 9 of MOSEK. Here is a preview implementation of some signal-to-interference-and-noise optimization problems in Python Fusion:

The key log-sum-exp constraint is just a few lines:

You can read more about modeling GPs in our Modeling Cookbook:

MOSEK's previous GP interface (scopt/dgopt/expopt), which is being phased out, relied on the slower and less accurate general nonlinear optimizer. You can read more about version 9 plans in

Tuesday, November 6, 2018

Reseller in China (Cardinal Operations) is the official reseller of MOSEK in China. Customers in China interested in acquiring a MOSEK license are welcome to visit

for details.

Friday, August 31, 2018

Conic modeling cheatsheet

As a supplement to the MOSEK Modeling Cookbook, here is s a quick reference guide to some useful conic models (click image for PDF):

Friday, August 17, 2018

Solving SDP with millions of matrix variables

Is it feasible in practice to solve semidefinite optimization problems with a huge number of matrix variables?

We recently received a problem from a structural engineering application with approximately the following  parameters:

  • 1 500 000 three-dimensional matrix variables,
  • 750 000 three-dimensional rotated quadratic cones,
  • $8\cdot 10^6$ scalar variables, $15\cdot 10^6$ linear constraints, $45\cdot 10^6$ nonzeros.

On a DELL PowerEdge R730 server with 2 Xeon E5-2687W v4 3.0GHZ the optimal solution is found in about 161 minutes on 24 threads using the latest MOSEK with memory peaking at about 60GB.  Due to the nature of the problem we disabled the linear dependency check and otherwise used all standard parameter settings.

Friday, June 29, 2018

.NET Core support

From release we initialize support for .NET Core, the cross-platform implementation of .NET.

MOSEK for .NET Core is distributed as a platform-independent NuGet package, which can be downloaded directly from our website. See the documentation for .NET APIs for installation instructions.

Friday, June 22, 2018

MOSEK at ISMP 2018

Here is a complete schedule of our talks at ISMP 2018 in chronological order:

Sven WieseThe Mixed-integer Conic Optimizer in MOSEKMixed-Integer Conic OptimizationMon, 5:00PMA. DURKHEIM, Bld. A
Henrik FribergProjection and presolve in MOSEK: exponential and power conesTheory and algorithms in conic linear programming 1Tue, 8:30AMS. LC5, Bld. L
Joachim DahlExtending MOSEK with exponential conesTheory and algorithms in conic linear programming 2Wed, 8:30AMS. 20, Bld. G
Erling AndersenMOSEK version 9Progress in Conic and MIP SolversWed, 3:15PMA. PITRES, Bld. O
MichaƂ AdamaszekExponential cone in MOSEK: overview and applicationsRelative Entropy Optimization IFri, 3:15PMS. LC5, Bld. L

Check the program for any last-minute changes.

The session on Mixed-Integer Conic Optimization (Mon, 5:00PM) is organized by Sven Wiese. The full program of that session is:

  • Lucas Letocart,  Exact methods based on SDP for the k-item quadratic knapsack problem
  • Tristan Gally, Knapsack Constraints over the Positive Semidefinite Cone
  • Sven Wiese, The Mixed-integer Conic Optimizer in MOSEK