20,441 research outputs found
A variational approach for continuous supply chain networks
We consider a continuous supply chain network consisting of buffering queues and processors first proposed by [D. Armbruster, P. Degond, and C. Ringhofer, SIAM J. Appl. Math., 66 (2006), pp. 896–920] and subsequently analyzed by [D. Armbruster, P. Degond, and C. Ringhofer, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), pp. 433–460] and [D. Armbruster, C. De Beer, M. Fre- itag, T. Jagalski, and C. Ringhofer, Phys. A, 363 (2006), pp. 104–114]. A model was proposed for such a network by [S. G ̈ottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559] using a system of coupling ordinary differential equations and partial differential equations. In this article, we propose an alternative approach based on a variational method to formulate the network dynamics. We also derive, based on the variational method, a computational algorithm that guarantees numerical stability, allows for rigorous error estimates, and facilitates efficient computations. A class of network flow optimization problems are formulated as mixed integer programs (MIPs). The proposed numerical algorithm and the corresponding MIP are compared theoretically and numerically with existing ones [A. Fu ̈genschuh, S. Go ̈ttlich, M. Herty, A. Klar, and A. Martin, SIAM J. Sci. Comput., 30 (2008), pp. 1490–1507; S. Go ̈ttlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559], which demonstrates the modeling and computational advantages of the variational approach
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
Power and Channel Allocation for Non-orthogonal Multiple Access in 5G Systems: Tractability and Computation
Network capacity calls for significant increase for 5G cellular systems. A
promising multi-user access scheme, non-orthogonal multiple access (NOMA) with
successive interference cancellation (SIC), is currently under consideration.
In NOMA, spectrum efficiency is improved by allowing more than one user to
simultaneously access the same frequency-time resource and separating
multi-user signals by SIC at the receiver. These render resource allocation and
optimization in NOMA different from orthogonal multiple access in 4G. In this
paper, we provide theoretical insights and algorithmic solutions to jointly
optimize power and channel allocation in NOMA. For utility maximization, we
mathematically formulate NOMA resource allocation problems. We characterize and
analyze the problems' tractability under a range of constraints and utility
functions. For tractable cases, we provide polynomial-time solutions for global
optimality. For intractable cases, we prove the NP-hardness and propose an
algorithmic framework combining Lagrangian duality and dynamic programming
(LDDP) to deliver near-optimal solutions. To gauge the performance of the
obtained solutions, we also provide optimality bounds on the global optimum.
Numerical results demonstrate that the proposed algorithmic solution can
significantly improve the system performance in both throughput and fairness
over orthogonal multiple access as well as over a previous NOMA resource
allocation scheme.Comment: IEEE Transactions on Wireless Communications, revisio
Combinatorial Continuous Maximal Flows
Maximum flow (and minimum cut) algorithms have had a strong impact on
computer vision. In particular, graph cuts algorithms provide a mechanism for
the discrete optimization of an energy functional which has been used in a
variety of applications such as image segmentation, stereo, image stitching and
texture synthesis. Algorithms based on the classical formulation of max-flow
defined on a graph are known to exhibit metrication artefacts in the solution.
Therefore, a recent trend has been to instead employ a spatially continuous
maximum flow (or the dual min-cut problem) in these same applications to
produce solutions with no metrication errors. However, known fast continuous
max-flow algorithms have no stopping criteria or have not been proved to
converge. In this work, we revisit the continuous max-flow problem and show
that the analogous discrete formulation is different from the classical
max-flow problem. We then apply an appropriate combinatorial optimization
technique to this combinatorial continuous max-flow CCMF problem to find a
null-divergence solution that exhibits no metrication artefacts and may be
solved exactly by a fast, efficient algorithm with provable convergence.
Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the
fact, already proved by Nozawa in the continuous setting, that the max-flow and
the total variation problems are not always equivalent.Comment: 26 page
Single-machine scheduling with stepwise tardiness costs and release times
We study a scheduling problem that belongs to the yard operations component of the railroad planning problems, namely the hump sequencing problem. The scheduling problem is characterized as a single-machine problem with stepwise tardiness cost objectives. This is a new scheduling criterion which is also relevant in the context of traditional machine scheduling problems. We produce complexity results that characterize some cases of the problem as pseudo-polynomially solvable. For the difficult-to-solve cases of the problem, we develop mathematical programming formulations, and propose heuristic algorithms. We test the formulations and heuristic algorithms on randomly generated single-machine scheduling problems and real-life datasets for the hump sequencing problem. Our experiments show promising results for both sets of problems
Energy-Efficient Scheduling for Homogeneous Multiprocessor Systems
We present a number of novel algorithms, based on mathematical optimization
formulations, in order to solve a homogeneous multiprocessor scheduling
problem, while minimizing the total energy consumption. In particular, for a
system with a discrete speed set, we propose solving a tractable linear
program. Our formulations are based on a fluid model and a global scheduling
scheme, i.e. tasks are allowed to migrate between processors. The new methods
are compared with three global energy/feasibility optimal workload allocation
formulations. Simulation results illustrate that our methods achieve both
feasibility and energy optimality and outperform existing methods for
constrained deadline tasksets. Specifically, the results provided by our
algorithm can achieve up to an 80% saving compared to an algorithm without a
frequency scaling scheme and up to 70% saving compared to a constant frequency
scaling scheme for some simulated tasksets. Another benefit is that our
algorithms can solve the scheduling problem in one step instead of using a
recursive scheme. Moreover, our formulations can solve a more general class of
scheduling problems, i.e. any periodic real-time taskset with arbitrary
deadline. Lastly, our algorithms can be applied to both online and offline
scheduling schemes.Comment: Corrected typos: definition of J_i in Section 2.1; (3b)-(3c);
definition of \Phi_A and \Phi_D in paragraph after (6b). Previous equations
were correct only for special case of p_i=d_
Column generation approaches to ship scheduling with flexible cargo sizes
We present a Dantzig-Wolfe procedure for the ship scheduling problem with flexible cargo sizes. This problem is similar to the well-known pickup and delivery problem with time windows, but the cargo sizes are defined by an interval instead of a fixed value. We show that the introduction of flexible cargo sizes to the column generation framework is not straightforward, and we handle the flexible cargo sizes heuristically when solving the subproblems. This leads to convergence issues in the branch-and-price search tree, and the optimal solution cannot be guaranteed. Hence we have introduced a method that generates an upper bound on the optimal objective. We have compared our method with an a priori column generation approach, and our computational experiments on real world cases show that the Dantzig-Wolfe approach is faster than the a priori generation of columns, and we are able to deal with larger or more loosely constrained instances. By using the techniques introduced in this paper, a more extensive set of real world cases can be solved either to optimality or within a small deviation from optimalityTransportation; integer programming; dynamic programming
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