Happy Pi Day

Code:

from mosek.fusion import *
from math import sqrt

def pi(n):
  M = Model()
  x = M.variable(n, Domain.inQCone())
  M.constraint(Expr.mulElm( range(1,n), x.slice(1,n) ),
               Domain.greaterThan(sqrt(6)))
  M.objective(ObjectiveSense.Minimize, x.index(0))
  M.solve()
  return M.primalObjValue()

print pi(100)
print pi(1000)
print pi(10000)
print pi(100000)
print pi(200000)


Output:

3.13198074852
3.14063710035
3.14149715507
3.14158310484
3.14158824714


Hint: 

$\sum_{i=1}^\infty\frac{1}{i^2}=\frac{\pi^2}{6}$.

MATLAB Computational Finance Conference 2018

MOSEK is one of the exhibitors at the MATLAB Computational Finance Conference 2018 in London on the 24th May. 

The event will include keynote presentations from The Bank of England and MathWorks as well as sessions with leading financial practitioners and academic institutions.

For details, speakers and registration see

https://www.mathworks.co.uk/matlabfinance

ISMP 2018

MOSEK is one of the sponsors of the 23rd International Symposium on Mathematical Programming ISMP 2018 which takes place in Bordeaux on 1-6 July 2018.

There will be plenty of opportunities to meet us in Bordeaux. Here are our tentative talk titles:

• Erling Andersen, MOSEK version 9
• Joachim Dahl, Extending MOSEK with exponential cones
• Henrik A. Friberg, Projection and presolve in MOSEK: exponential and power cones
• Sven Wiese, Mixed-integer Conic Optimization in MOSEK in a session on Mixed-integer Conic Optimization Sven is organizing.
• Michał Adamaszek, Exponential cone in MOSEK: overview and applications

Stay tuned for more details closer to the date.

Modern Optimization in Energy - Summer School

DTU CEE Summer School 2018 Modern Optimization in Energy will be held June 24-29, 2018 at the Technical University of Denmark. This is the third edition of the summer school. See list of speakers, program and how to apply (until March 18th).

Just like last year MOSEK is one of the sponsors of the meeting and of two MOSEK scholarships for outstanding student participants.

Version 9 roadmap

MOSEK version 9 is scheduled for release in early 2019. In this blog post we will outline some of the expected changes. Please note this is subject to change and hence the blogpost will be updated occasionally.

The major new feature in version 9 is support for exponential cone

$$x\geq y\exp(z/y)$$

and power cone

$$x^\alpha y^{1-\alpha}\geq |z|$$

This has the implication that almost any practical convex optimization problem can be formulated directly on conic form. That includes modeling functions such as $x^\alpha$, $x^{1.5}$, $\log{x}$, $e^x$, relative entropy, $p$-norm, logistic, Kullback–Leibler divergence, Cobb-Douglas etc.

Mixed-integer optimization will be available for all conic problems without semi-definite matrix variables. Hence, version 9 should be able to handle almost all mixed-integer optimization problems appearing in practice.

An advantage of moving to the conic form is that it become possible for MOSEK to solve the dual problem if deemed worthwhile. This sometimes leads to dramatic speed ups.

The general nonlinear convex optimizer will be dropped, since optimization problems in conic form are preferred for efficiency and stability reasons. Quadratic and quadratically constrained problems remain unaffected.

Below the planned changes are listed in some detail. Please send us an e-mail to support@mosek.com if you have questions or comments.

Feature changes

• Adding support for the exponential and power cones in all interfaces.
• Adding mixed-integer support for new cone types.
• Dropping support for general convex optimization problems.

Interface changes

OPTIMIZER API

• Remove SCopt, DGopt, EXPopt which depended on the general nonlinear optimizer. 

R

• Remove scopt which depended on the general nonlinear optimizer.
• Add support for conic constraints of the form $Fx+g\in K$.

TOOLBOX FOR MATLAB

• Remove mskenopt, mskscopt and mskgpopt which depended on the general nonlinear optimizer.
• Add support for conic constraints of the form $Fx+g\in K$.

FUSION

• Fusion for MATLAB will be dropped due to technical issues and low demand. 
• Improved performance.

Other changes

• Vastly improved performance on the AMD Ryzen family of CPUs. 
• New version of the Modeling Cookbook adapted to new conic features.
• Dropped support reading and writing XML formatted files.  
• Dropped near-optimal, near-infeasible and other near- solution statuses.

Christmas 2017

Sales and support will be closed on 25,26 December and 1 January.

We wish everyone good holiday.

MOSEK Team

The Christmas trees were created using logistic regression with different levels of regularization. Our implementation used the exponential cone which will be introduced in MOSEK version 9, to be released in 2018. Stay tuned!

Workshop: Mixed-integer conic optimization

We are pleased to announce a MOSEK workshop on Mixed-integer conic optimization taking place on Thursday, January 11th, 2018 at our place in the Symbion research park, Copenhagen.

The workshop is free and open to everyone. There will be coffee, refreshments and time for discussions. Please register through this form to help us with planning.

Schedule:
14:00 - 14:05   Welcome (Erling Andersen)
14:05 - 14:50   Tristan Gally
15:00 - 15:45   Julio C. Góez
16:00 - 16:45   Joachim Dahl
17:30+ optional dinner (Nørrebro Bryghus)

Abstracts:

• Tristan Gally, TU Darmstadt, Applications and Solution Approaches for Mixed-Integer Semidefinite ProgrammingMixed-integer semidefinite programming (MISDP) has received increasing attention in recent years. MISDPs appear in many applications either by adding combinatorial decisions to nonlinear problems with natural SDP-formulations or by reformulating combinatorial optimization problems to incorporate stronger SDP-relaxations. While mixed-integer second-order cone programming has been adapted by many commercial solvers, MISDP remains a challenging problem, which so far has mostly been tackled by solution-specific approaches.
  
  In this talk, we want to present some interesting applications for MISDP from both combinatorial and nonlinear optimization. Afterwards, we will discuss problem-independent solution approaches, mainly concentrating on nonlinear branch-and-bound. Particularly, we will explain the importance of the Slater constraint qualification and its implications for using interior-point methods within a branch-and-bound approach. We will further discuss enhancing techniques like dual fixing and warmstarts and give numerical results comparing the different solution approaches and different implementations.
• Julio C. Góez, NHH, Disjunctive conic cuts: the good, the bad, and implementation In recent years, the generalization of Balas disjunctive cuts for mixed integer linear optimization problems to mixed integer non-linear optimization problems has received significant attention. Among these studies, mixed integer second order cone optimization (MISOCO) is a special case. For MISOCO one has the disjunctive conic cuts approach. That generalization introduced the concept of disjunctive conic cuts (DCCs) and disjunctive cylindrical cuts (DCyCs). Specifically, it showed that under some mild assumptions the intersection of those DCCs and DCyCs with a closed convex set, given as the intersection of a second order cone and an affine set, is the convex hull of the intersection of the same set with a parallel linear disjunction. The key element in that analysis was the use of pencils of quadrics to find close forms for deriving the DCCs and DCyCs. The first part of this talk will summarize the main results about DCCs and DCyCs including some results about valid conic inequalities for hyperboloids and non-convex quadratic cones when the disjunction is defined by parallel hyperplanes. In the second part, we will discussed some of the limitation of this approach to derive useful valid inequalities in the context of MISOCO. In the last part, we will briefly describe the software libraries that together constitute DisCO, a full-featured solver for MISOCP which we are currently used to explore the potential of DCCs and DCyCs.
• Joachim Dahl, MOSEK, Mixed-integer conic optimization using MOSEK
  Recently Lubin et.al. showed that all the convex instances of the nonlinear mixed-integer bench library MINLPLIB2 can be reformulated as conic optimization problems using 5 different cone types which are the linear, the quadratic, the semidefinite, the exponential and the power cones. The former three cones belong to the class of symmetric cones whereas the latter two belong to the class of nonsymmetric cones. We call modelling with affine expressions and the five previously mentioned cone types extreme desciplined modelling. Based on Lubin et al. and the experience at our company we claim almost all practical convex optimization can be expressed using extreme disciplined modelling so it is a general framework. Now it is much easier to build optimization algorithms and software for extreme disciplined optimization models rather than for general convex (unstructured) convex problems because of limited and explicit structure. This fact is exploited in the software package MOSEK to be discussed. The software package MOSEK has for many years been able to solve conic optimization over the symmetric cones but in the upcoming version 9 MOSEK can also handle two nonsymmetric cones i.e. the exponential and the power cone. In this presentation we will discuss the continuous and mixed-integer conic optimizer in MOSEK. In addition extensive computational results are presented that illustrate the performance of MOSEK on problems including nonsymmetric cones.

Essentials:

• Time: 11.01.2018, from 2pm.
• Place: Fruebjergvej 3, 2100 Copenhagen, directions to the Symbion research park.
• Registration: use this form
• Contact: info@mosek.com