4,341 research outputs found
Slow Adaptive OFDMA Systems Through Chance Constrained Programming
Adaptive OFDMA has recently been recognized as a promising technique for
providing high spectral efficiency in future broadband wireless systems. The
research over the last decade on adaptive OFDMA systems has focused on adapting
the allocation of radio resources, such as subcarriers and power, to the
instantaneous channel conditions of all users. However, such "fast" adaptation
requires high computational complexity and excessive signaling overhead. This
hinders the deployment of adaptive OFDMA systems worldwide. This paper proposes
a slow adaptive OFDMA scheme, in which the subcarrier allocation is updated on
a much slower timescale than that of the fluctuation of instantaneous channel
conditions. Meanwhile, the data rate requirements of individual users are
accommodated on the fast timescale with high probability, thereby meeting the
requirements except occasional outage. Such an objective has a natural chance
constrained programming formulation, which is known to be intractable. To
circumvent this difficulty, we formulate safe tractable constraints for the
problem based on recent advances in chance constrained programming. We then
develop a polynomial-time algorithm for computing an optimal solution to the
reformulated problem. Our results show that the proposed slow adaptation scheme
drastically reduces both computational cost and control signaling overhead when
compared with the conventional fast adaptive OFDMA. Our work can be viewed as
an initial attempt to apply the chance constrained programming methodology to
wireless system designs. Given that most wireless systems can tolerate an
occasional dip in the quality of service, we hope that the proposed methodology
will find further applications in wireless communications
Random projections for linear programming
Random projections are random linear maps, sampled from appropriate
distributions, that approx- imately preserve certain geometrical invariants so
that the approximation improves as the dimension of the space grows. The
well-known Johnson-Lindenstrauss lemma states that there are random ma- trices
with surprisingly few rows that approximately preserve pairwise Euclidean
distances among a set of points. This is commonly used to speed up algorithms
based on Euclidean distances. We prove that these matrices also preserve other
quantities, such as the distance to a cone. We exploit this result to devise a
probabilistic algorithm to solve linear programs approximately. We show that
this algorithm can approximately solve very large randomly generated LP
instances. We also showcase its application to an error correction coding
problem.Comment: 26 pages, 1 figur
The CONEstrip algorithm
Uncertainty models such as sets of desirable gambles and (conditional) lower previsions can be represented as convex cones. Checking the consistency of and drawing inferences from such models requires solving feasibility and optimization problems. We consider finitely generated such models. For closed cones, we can use linear programming; for conditional lower prevision-based cones, there is an efficient algorithm using an iteration of linear programs. We present an efficient algorithm for general cones that also uses an iteration of linear programs
Probabilistic analysis of a differential equation for linear programming
In this paper we address the complexity of solving linear programming
problems with a set of differential equations that converge to a fixed point
that represents the optimal solution. Assuming a probabilistic model, where the
inputs are i.i.d. Gaussian variables, we compute the distribution of the
convergence rate to the attracting fixed point. Using the framework of Random
Matrix Theory, we derive a simple expression for this distribution in the
asymptotic limit of large problem size. In this limit, we find that the
distribution of the convergence rate is a scaling function, namely it is a
function of one variable that is a combination of three parameters: the number
of variables, the number of constraints and the convergence rate, rather than a
function of these parameters separately. We also estimate numerically the
distribution of computation times, namely the time required to reach a vicinity
of the attracting fixed point, and find that it is also a scaling function.
Using the problem size dependence of the distribution functions, we derive high
probability bounds on the convergence rates and on the computation times.Comment: 1+37 pages, latex, 5 eps figures. Version accepted for publication in
the Journal of Complexity. Changes made: Presentation reorganized for
clarity, expanded discussion of measure of complexity in the non-asymptotic
regime (added a new section
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