Our support and sales will be closed for Easter holidays from and including Thursday, April 13th until and including Monday, April 17th. We resume on Tuesday, April 18th.
Wishing everyone a good Easter,
The MOSEK team
Tuesday, April 4, 2017
DTU Power Systems and Electricity Markets School 2017
On June 12-16 the The Energy Analytics and Markets Group at the Technical University of Denmark (DTU) is hosting a Summer School Modern Challenges in Power System Operation and Electricity Markets: An Optimization Perspective. This is the second DTU summer school on the topic of electricity markets and power systems, and as before it will have outstanding speakers.
We are very happy to be one of the sponsors of the school. In particular, two MOSEK scholarships are waiting for two outstanding student participants.
Registration is open until May 7th. We hope to see you at the school in June!
Friday, March 3, 2017
Power flow problems - workshop summary
On February 28th we held the Workshop on Semidefinite Optimization in Power Flow problems.
- Spyros Chatzivasileiadis gave a talk about SDO methods for producing stability certificates for power systems and about the optimal power flow under uncertainty.
- Cédric Josz introduced the complex variant of the Lasserre moment hierarchy and discussed the possible advantages of a convex optimizer working directly over the complex numbers.
- Martin Skovgaard Andersen talked about numerical aspects and experiments with convex relaxations of optimal flow problems. In particular, he was able to solve the SDP relaxations of test cases with over 10K power buses using MOSEK.
The slides from all three talks can be found on our website.
We thank the speakers and the participants for making this a great workshop!
MOSEK Team
Friday, February 24, 2017
Sponsorship: NemFest, Atlanta, 11-12 May 2017
We are very proud to be one of the sponsors of NemFest 2017 taking place May 11-12, 2017 in Atlanta.
NemFest 2017 is a conference in honor of two extraordinary researchers who shaped the area of discrete and comtinuous optimization: George Nemhauser and Arkadi Nemirovski.
Arkadi Nemirovski was one of the first users of MOSEK back in 1998, even before the release of the first official version.
NemFest 2017 is a conference in honor of two extraordinary researchers who shaped the area of discrete and comtinuous optimization: George Nemhauser and Arkadi Nemirovski.
Arkadi Nemirovski was one of the first users of MOSEK back in 1998, even before the release of the first official version.
MOSEK Team
Tuesday, February 21, 2017
Sponsorship: MIP 2017 - Montréal, 19-22 June
MOSEK is once again one of the sponsors of the Mixed Integer Programming Workshop, which will take place June 19-22, 2017 at HEC Montréal (Québec, Canada).
The program includes talks from distinguished specialists from the academic and industrial world. Until March 1st you can still submit a poster abstract and apply for travel support for students and postdocs.
More details on the official workshop website.
MOSEK Team
Tuesday, January 3, 2017
Semidefinite optimization in power flow problems
We are pleased to announce a MOSEK workshop on Semidefinite optimization in power flow problems taking place on Tuesday, February 28th, 2017 at the Symbion research park.
Optimal power flow is one of the major problems in optimization of electric power systems, asking for the minimization of operating costs in terms of a specified objective function in the presence of non-linear power flow equations. Three experts, Spyros Chatzivasileiadis (DTU), Cédric Josz (CNRS) and Martin Skovgaard Andersen (DTU) will discuss recent advanced based on convex relaxations and in particular on semidefinite programming.
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
14:05 - 14:50 Spyros Chatzivasileiadis
15:00 - 15:45 Cédric Josz
16:00 - 16:45 Martin Skovgaard Andersen
17:30+ optional dinner (Nørrebro Bryghus)
Abstracts:
- Spyros Chatzivasileiadis, DTU
SDP Problems for Power System Stability and Optimization
In recent years, semidefinite programming is met with increasing interest within the power systems community. Its most notable application to-date is on the convex formulation of the AC optimal power flow problem. At the same time, semidefinite programs can be used to derive Lyapunov functions that guarantee power system stability.
In this talk we will report on recent work both on power system stability and optimization. First, we will present a novel robust stability toolbox for power grids that can address uncertainties in equilibrium points and fault-on dynamics. In that, we bring in the quadratic Lyapunov functions approach to transient stability assessment.
Second, we will propose formulations for the integration of chance constraints for several uncertain variables in the optimal power flow problem. We demonstrate our method with numerical examples, and we investigate the conditions to achieve zero duality gap. - Cédric Josz, LAAS CNRS
Application of Polynomial Optimization to Electricity Transmission Networks
Multivariate polynomial optimization where variables and data are complex numbers is a non-deterministic polynomial-time hard problem that arises in various applications such as electric power systems, imaging science, signal processing, and quantum mechanics. We transpose to complex numbers the Lasserre hierarchy which aims to solve real polynomial optimization problems to global optimality. This brings complex semidefinite programming into the picture and calls for an interior-point algorithm in complex numbers. The Nesterov-Todd direction will be discussed and supplemented by numerical results on the European high-voltage electricity transmission network. - Martin Skovgaard Andersen, DTU
Numerical Aspects of Semidefinite Relaxations of Optimal Power Flow Problems
Power flow optimization plays an important role in power system operation and planning. It is used to find a cost-optimal operating point of a power system that consists of a set of power buses that are interconnected through a network of transmission lines. We discuss recent progress based on convex relaxation techniques for optimal power flow problems and investigate some numerical aspects through an empirical study.
Essentials:
- Time: 28.02.2017, from 2pm.
- Place: Fruebjergvej 3, 2100 Copenhagen, directions to the Symbion research park.
- Registration: use this form
- Contact: info@mosek.com
Thursday, December 1, 2016
Modeling non-convex absolute value constraints with MILP
We are being asked if one can use MOSEK to express a constraint of the form $$\label{eq:sumabs}\sum_i|x_i|=c$$ for some $x\in\mathbb{R}^n$ and $c\in\mathbb{R}$. This condition is non-convex; for instance for $n=1$ it is equivalent to $x=\pm c$. Such constraints appear in long-short portfolio optimization problems (see stackoverflow, paper).
The idea is to introduce a binary vector indicating the positions of positive/negative entries in $x$. Concretely, we want to split $x$ into the positive and negative part $x=s-t$ with $s,t\geq 0$.
If we know an upper bound $M$ on $s_i$ and $t_i$, then the following system:
$$\begin{align*}
0\leq s_i&\leq My_i,\\
0\leq t_i&\leq M(1-y_i),\\
& y_i\in \{0,1\}
\end{align*}$$ has the property that at most one of $s_i,t_i$ is non-zero. Indeed:
$$
\begin{align*}
s_i>0 \implies y_i=1 \implies t_i=0,\\
t_i>0 \implies y_i=0 \implies s_i=0.
\end{align*}
$$ Given that we are looking at $\sum_i|x_i|=c$ we can choose $M=c$ as an upper bound for all $|x_i|$. Then an equivalent version of the condition $\eqref{eq:sumabs}$ as a mixed-integer linear program is:
$$
\begin{align*}
x&=s-t,\\
0\leq s&\leq cy,\\
0\leq t&\leq c(1-y),\\
0\leq y&\leq 1,\\
c&=\sum_is_i+\sum_it_i,\\
& x,s,t\in\mathbb{R}^n,\ y\in\mathbb{Z}^n.\\
\end{align*}
$$
The idea is to introduce a binary vector indicating the positions of positive/negative entries in $x$. Concretely, we want to split $x$ into the positive and negative part $x=s-t$ with $s,t\geq 0$.
If we know an upper bound $M$ on $s_i$ and $t_i$, then the following system:
$$\begin{align*}
0\leq s_i&\leq My_i,\\
0\leq t_i&\leq M(1-y_i),\\
& y_i\in \{0,1\}
\end{align*}$$ has the property that at most one of $s_i,t_i$ is non-zero. Indeed:
$$
\begin{align*}
s_i>0 \implies y_i=1 \implies t_i=0,\\
t_i>0 \implies y_i=0 \implies s_i=0.
\end{align*}
$$ Given that we are looking at $\sum_i|x_i|=c$ we can choose $M=c$ as an upper bound for all $|x_i|$. Then an equivalent version of the condition $\eqref{eq:sumabs}$ as a mixed-integer linear program is:
$$
\begin{align*}
x&=s-t,\\
0\leq s&\leq cy,\\
0\leq t&\leq c(1-y),\\
0\leq y&\leq 1,\\
c&=\sum_is_i+\sum_it_i,\\
& x,s,t\in\mathbb{R}^n,\ y\in\mathbb{Z}^n.\\
\end{align*}
$$
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