5,956 research outputs found
On the Hardness of Entropy Minimization and Related Problems
We investigate certain optimization problems for Shannon information
measures, namely, minimization of joint and conditional entropies ,
, , and maximization of mutual information , over
convex regions. When restricted to the so-called transportation polytopes (sets
of distributions with fixed marginals), very simple proofs of NP-hardness are
obtained for these problems because in that case they are all equivalent, and
their connection to the well-known \textsc{Subset sum} and \textsc{Partition}
problems is revealed. The computational intractability of the more general
problems over arbitrary polytopes is then a simple consequence. Further, a
simple class of polytopes is shown over which the above problems are not
equivalent and their complexity differs sharply, namely, minimization of
and is trivial, while minimization of and
maximization of are strongly NP-hard problems. Finally, two new
(pseudo)metrics on the space of discrete probability distributions are
introduced, based on the so-called variation of information quantity, and
NP-hardness of their computation is shown.Comment: IEEE Information Theory Workshop (ITW) 201
The N-K Problem in Power Grids: New Models, Formulations and Numerical Experiments (extended version)
Given a power grid modeled by a network together with equations describing
the power flows, power generation and consumption, and the laws of physics, the
so-called N-k problem asks whether there exists a set of k or fewer arcs whose
removal will cause the system to fail. The case where k is small is of
practical interest. We present theoretical and computational results involving
a mixed-integer model and a continuous nonlinear model related to this
question.Comment: 40 pages 3 figure
Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation
We consider a utility maximization problem for an investment-consumption
portfolio when the current utility depends also on the wealth process. Such
kind of problems arise, e.g., in portfolio optimization with random horizon or
with random trading times. To overcome the difficulties of the problem we use
the dual approach. We define a dual problem and treat it by means of dynamic
programming, showing that the viscosity solutions of the associated
Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth
functions. This allows to define a smooth solution of the primal
Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique
in a suitable class and coincides with the value function of the primal
problem. Some financial applications of the results are provided
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