4,628 research outputs found
Dynamic Energy Management
We present a unified method, based on convex optimization, for managing the
power produced and consumed by a network of devices over time. We start with
the simple setting of optimizing power flows in a static network, and then
proceed to the case of optimizing dynamic power flows, i.e., power flows that
change with time over a horizon. We leverage this to develop a real-time
control strategy, model predictive control, which at each time step solves a
dynamic power flow optimization problem, using forecasts of future quantities
such as demands, capacities, or prices, to choose the current power flow
values. Finally, we consider a useful extension of model predictive control
that explicitly accounts for uncertainty in the forecasts. We mirror our
framework with an object-oriented software implementation, an open-source
Python library for planning and controlling power flows at any scale. We
demonstrate our method with various examples. Appendices give more detail about
the package, and describe some basic but very effective methods for
constructing forecasts from historical data.Comment: 63 pages, 15 figures, accompanying open source librar
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Asset Allocation under the Basel Accord Risk Measures
Financial institutions are currently required to meet more stringent capital
requirements than they were before the recent financial crisis; in particular,
the capital requirement for a large bank's trading book under the Basel 2.5
Accord more than doubles that under the Basel II Accord. The significant
increase in capital requirements renders it necessary for banks to take into
account the constraint of capital requirement when they make asset allocation
decisions. In this paper, we propose a new asset allocation model that
incorporates the regulatory capital requirements under both the Basel 2.5
Accord, which is currently in effect, and the Basel III Accord, which was
recently proposed and is currently under discussion. We propose an unified
algorithm based on the alternating direction augmented Lagrangian method to
solve the model; we also establish the first-order optimality of the limit
points of the sequence generated by the algorithm under some mild conditions.
The algorithm is simple and easy to implement; each step of the algorithm
consists of solving convex quadratic programming or one-dimensional
subproblems. Numerical experiments on simulated and real market data show that
the algorithm compares favorably with other existing methods, especially in
cases in which the model is non-convex
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