11,254 research outputs found
A probabilistic tree-based representation for non-convex minimum cost flow problems
Network flow optimisation has many real-world applications. The minimum cost flow problem (MCFP) is one of the most common network flow problems. Mathematical programming methods often assume the linearity and convexity of the underlying cost function, which is not realistic in many real-world situations. Solving large-sized MCFPs with nonlinear non-convex cost functions poses a much harder problem. In this paper, we propose a new representation scheme for solving non-convex MCFPs using genetic algorithms (GAs). The most common representation scheme for solving the MCFP in the literature using a GA is priority-based encoding, but it has some serious limitations including restricting the search space to a small part of the feasible set. We introduce a probabilistic tree-based representation scheme (PTbR) that is far superior compared to the priority-based encoding. Our extensive experimental investigations show the advantage of our encoding compared to previous methods for a variety of cost functions
Graphical Models for Optimal Power Flow
Optimal power flow (OPF) is the central optimization problem in electric
power grids. Although solved routinely in the course of power grid operations,
it is known to be strongly NP-hard in general, and weakly NP-hard over tree
networks. In this paper, we formulate the optimal power flow problem over tree
networks as an inference problem over a tree-structured graphical model where
the nodal variables are low-dimensional vectors. We adapt the standard dynamic
programming algorithm for inference over a tree-structured graphical model to
the OPF problem. Combining this with an interval discretization of the nodal
variables, we develop an approximation algorithm for the OPF problem. Further,
we use techniques from constraint programming (CP) to perform interval
computations and adaptive bound propagation to obtain practically efficient
algorithms. Compared to previous algorithms that solve OPF with optimality
guarantees using convex relaxations, our approach is able to work for arbitrary
distribution networks and handle mixed-integer optimization problems. Further,
it can be implemented in a distributed message-passing fashion that is scalable
and is suitable for "smart grid" applications like control of distributed
energy resources. We evaluate our technique numerically on several benchmark
networks and show that practical OPF problems can be solved effectively using
this approach.Comment: To appear in Proceedings of the 22nd International Conference on
Principles and Practice of Constraint Programming (CP 2016
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Risk-Averse Model Predictive Operation Control of Islanded Microgrids
In this paper we present a risk-averse model predictive control (MPC) scheme
for the operation of islanded microgrids with very high share of renewable
energy sources. The proposed scheme mitigates the effect of errors in the
determination of the probability distribution of renewable infeed and load.
This allows to use less complex and less accurate forecasting methods and to
formulate low-dimensional scenario-based optimisation problems which are
suitable for control applications. Additionally, the designer may trade
performance for safety by interpolating between the conventional stochastic and
worst-case MPC formulations. The presented risk-averse MPC problem is
formulated as a mixed-integer quadratically-constrained quadratic problem and
its favourable characteristics are demonstrated in a case study. This includes
a sensitivity analysis that illustrates the robustness to load and renewable
power prediction errors
Power packet transferability via symbol propagation matrix
Power packet is a unit of electric power transferred by a power pulse with an
information tag. In Shannon's information theory, messages are represented by
symbol sequences in a digitized manner. Referring to this formulation, we
define symbols in power packetization as a minimum unit of power transferred by
a tagged pulse. Here, power is digitized and quantized. In this paper, we
consider packetized power in networks for a finite duration, giving symbols and
their energies to the networks. A network structure is defined using a graph
whose nodes represent routers, sources, and destinations. First, we introduce
symbol propagation matrix (SPM) in which symbols are transferred at links
during unit times. Packetized power is described as a network flow in a
spatio-temporal structure. Then, we study the problem of selecting an SPM in
terms of transferability, that is, the possibility to represent given energies
at sources and destinations during the finite duration. To select an SPM, we
consider a network flow problem of packetized power. The problem is formulated
as an M-convex submodular flow problem which is known as generalization of the
minimum cost flow problem and solvable. Finally, through examples, we verify
that this formulation provides reasonable packetized power.Comment: Submitted to Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Science
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