## Friday, June 24, 2022

### MOSEK 10 release approaching

MOSEK version 10 with all its new features has been available as a beta version for a while now. See https://www.mosek.com/products/version-10/ for the latest details and download and installation instructions.

Huge thanks to all the users who were so kind to test this beta release and share their suggestions, bug reports and comments. We are working on incorporating your input into the final release. Do not hesitate to write to us with further remarks.

MOSEK 10 is planned to become the current stable release latest on August 15th 2022.

In the meantime, we wish all MOSEK users a good summer, enjoyable vacation, and hope to meet you with version 10 after the summer.

The MOSEK team

## Wednesday, June 22, 2022

### On the performance improvement of the semi-definite optimizer when switching from version 9 to 10

The most reason Mosek version 10.0.16(beta) includes a very significant improvement of the optimizer for semi-definite problems. The improvement is most notable for problems having many matrix variables. In order to illustrate the potential performance gain we use a problem supplied by Mosek user. Some statistics about the problem can be seen in the Mosek log output:


Optimizer  - solved problem         : the primal
Optimizer  - Constraints            : 8808
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 0                 conic                  : 0
Optimizer  - Semi-definite variables: 49                scalarized             : 76493
Factor     - setup time             : 8.26              dense det. time        : 0.00
Factor     - ML order time          : 0.50              GP order time          : 0.06
Factor     - nonzeros before factor : 3.38e+07          after factor           : 3.38e+07
Factor     - dense dim.             : 0                 flops                  : 2.45e+11
Factor     - GP saved nzs           : 4.97e+06          GP saved flops         : 4.60e+10

Hence, the problem has 8808 constraints and 49 matrix variables. In addition it should be mentioned that the contraint matrix is not that sparse.

Below the computational results is shown.

CPU 9.3.20 9.3.20 (MT) 10.0.16 10.0.16 (MT)
Intel Xeon Gold 6126 (12 core) 1165 1171 833 207
AMD EPYC 7402P (24 core) 1189 1049 777 108
The table shows the computational time in wall clock seconds for running version 9.3.20 and 10.0.16 single and multi-threaded respectively. Moreover, the timing result is shown for an Intel and an AMD CPU.

The main conclusions from the table are

• version 10 is much faster than version 9 and the difference is more notable on the AMD CPU.
• version 10 offers much better speed-up when using multiple threads. In fact there are virtually no speed-up when using multiple threads in version 9.
It should emphasized that the performance improvement for solving semi-definite problems when switching from version 9 to version 10 is dependent on the structure of the problem. However, as illustrated above a big performance boost can be realized for some problems.

PS. Despite the AMD CPU being the fastest in the table above then this post do not claim AMD CPUs are better than Intel CPUs or the other way around.

## Monday, May 23, 2022

### Symmetry detection for MIPs in MOSEK 10

Remember counting ladybug spots as a child? If you realized ladybugs were symmetric, you could save some work by counting on one wing only and multiplying by two. In Mixed Integer Programming, something similar applies. If we detect symmetries in a problem formulation, we may be able to save some computational effort and thus time when solving it. But first of all, what are symmetries in a MIP formulation? Different from the geometric symmetry of a ladybug, the notion of symmetries in MIP may be thought of as the presence of equivalent solutions.

They can arise, for example, when identical physical objects are described distinctively in a model. As an example, take the MISOCP models for geometric facility location problems described here. Roughly speaking, the task there is to cover a set of given points in the plane with disks to be placed at will. Say we have 10 points and 3 fixed-radius disks, and want to cover as many points as possible. In a straightforward model, we would assign indexes $i = 1,2,3$ to the disks, and indexes $j = 1,...,10$ to the points, and then define all sorts of variables like $x_{ij} \in \{0,1\}$, stating whether disk $i$ covers point $j$. What we did here is assign labels to the disks, making them a blue, a red and a green one, say. But if all three disks have the same radius, this gives rise to equivalent solutions, see Figure 1. For the cost of a solution, it is only important whether a certain point is covered, but not by which disk.

Figure 1: Equivalent disk covering solutions

Symmetry breaking inequalities, orbital branching and fixing, or Isomorphism pruning are just some of the various techniques that have been studied over the years to better address symmetries in MIP. Variants of some of these various techniques have been implemented in the latest MOSEK version 10. Figure 2 sketches the effect on solution times of completely switching off symmetry detection and handling with the user parameter MSK_IPAR_MIO_SYMMETRY_LEVEL on the so-called Margot instances, known to be highly symmetric.
Figure 2: Solving Margot's instances w/ and w/o symmetry detection

Some of the techniques in question can be extended from Mixed Integer Linear to Mixed Integer Conic Programming, or more generally Mixed Integer Nonlinear Programming. We sometimes advocate that one nice feature of (Mixed Integer) Conic Programming is the separation of data and structure, and implementing symmetry detection is an aspect where this comes in handy. All data involved in a model is contained in linear constraints like $a^Tx\leq b$, and the symmetry detection routines needed here are already needed in MILP. Now extending symmetry detection to conic constraints, e.g., to a conic-quadratic constraint $x_0\geq \Vert x_{1:n}\Vert_2$, is relatively easy because one does not have to hassle with data, but only structure. Adding symmetry detection for possibly newly added cone types to a MICP solver is thus a small burden. And detecting and exploting symmetry in MICP can have a decisive effect. The following logs show MOSEK's solution paths to solve one of the aforementioned geometric facility location problems, with and without symmetry detection and handling, respectively.

Figure 3: Solving disk covering instances w/ and w/o symmetry detection

## Tuesday, April 5, 2022

### Easter 2022

Due to Easter support and sales are closed from Thursday, April 14th, until Monday, April 18th, inclusive.

The MOSEK team wishes everyone a peaceful Easter.

## Friday, March 25, 2022

### Improved performance of the interior-point optimizer when using an AMD CPU

One of the improvements in the recent released Mosek version 10 is better performance when run on a computer that has a recent AMD CPU. In order to illustrate that the table below is shown.

 Problem 9.3 10.0 pds-100 93s 66s

The table shows the time taken in seconds to solve some well-known linear optimization problems on a AMD EPYC 7543 CPU from the Zen3 familie running the interior-point optimizer on 1 thread. On this AMD Zen3 family CPU Mosek version 10 is approximately 30% faster which is a significant improvement. It should be emphasized that on easier problems than pds-100 then the speed improvement is less than reported above. Nevertheless an upgrade to Mosek version 10 should be particularly worthwhile when using an AMD CPU.

## Wednesday, March 23, 2022

### MOSEK 10 Beta release

We are excited to announce the release of MOSEK 10 Beta.

This version introduces many new features, in particular:

• Support for Apple M1.
• Efficiency improvements in the interior-point and mixed-integer optimizers, improved tuning for various platforms.
• New strategies in the mixed-integer optimizer.
• Disjunctive constraints.
• More convenient syntax for conic constraints in the Optimizer API.
• And many more improvements.
To read more and try the new version see https://www.mosek.com/products/version-10/

In the coming weeks we will publish a series of posts highlighting selected new features of MOSEK 10.

Your feedback will be most appreciated.

The MOSEK team

## Thursday, January 20, 2022

### MIP 2022

Similarly to previous years MOSEK is proud to be one of the sponsors of MIP2022 - Mixed Integer Programming Workshop 2022, taking place May 23–26, 2022 at DIMACS, Rutgers University.

More details at https://www.mixedinteger.org/2022/