87 research outputs found
Characterizing Optimal Adword Auctions
We present a number of models for the adword auctions used for pricing
advertising slots on search engines such as Google, Yahoo! etc. We begin with a
general problem formulation which allows the privately known valuation per
click to be a function of both the identity of the advertiser and the slot. We
present a compact characterization of the set of all deterministic incentive
compatible direct mechanisms for this model. This new characterization allows
us to conclude that there are incentive compatible mechanisms for this auction
with a multi-dimensional type-space that are {\em not} affine maximizers. Next,
we discuss two interesting special cases: slot independent valuation and slot
independent valuation up to a privately known slot and zero thereafter. For
both of these special cases, we characterize revenue maximizing and efficiency
maximizing mechanisms and show that these mechanisms can be computed with a
worst case computational complexity and respectively,
where is number of bidders and is number of slots. Next, we
characterize optimal rank based allocation rules and propose a new mechanism
that we call the customized rank based allocation. We report the results of a
numerical study that compare the revenue and efficiency of the proposed
mechanisms. The numerical results suggest that customized rank-based allocation
rule is significantly superior to the rank-based allocation rules.Comment: 29 pages, work was presented at a) Second Workshop on Sponsored
Search Auctions, Ann Arbor, MI b) INFORMS Annual Meeting, Pittsburgh c)
Decision Sciences Seminar, Fuqua School of Business, Duke Universit
Exponential penalty function control of loss networks
We introduce penalty-function-based admission control policies to
approximately maximize the expected reward rate in a loss network. These
control policies are easy to implement and perform well both in the transient
period as well as in steady state. A major advantage of the penalty approach is
that it avoids solving the associated dynamic program. However, a disadvantage
of this approach is that it requires the capacity requested by individual
requests to be sufficiently small compared to total available capacity. We
first solve a related deterministic linear program (LP) and then translate an
optimal solution of the LP into an admission control policy for the loss
network via an exponential penalty function. We show that the penalty policy is
a target-tracking policy--it performs well because the optimal solution of the
LP is a good target. We demonstrate that the penalty approach can be extended
to track arbitrarily defined target sets. Results from preliminary simulation
studies are included.Comment: Published at http://dx.doi.org/10.1214/105051604000000936 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A First-order Augmented Lagrangian Method for Compressed Sensing
We propose a first-order augmented Lagrangian algorithm (FAL) for solving the
basis pursuit problem. FAL computes a solution to this problem by inexactly
solving a sequence of L1-regularized least squares sub-problems. These
sub-problems are solved using an infinite memory proximal gradient algorithm
wherein each update reduces to "shrinkage" or constrained "shrinkage". We show
that FAL converges to an optimal solution of the basis pursuit problem whenever
the solution is unique, which is the case with very high probability for
compressed sensing problems. We construct a parameter sequence such that the
corresponding FAL iterates are eps-feasible and eps-optimal for all eps>0
within O(log(1/eps)) FAL iterations. Moreover, FAL requires at most O(1/eps)
matrix-vector multiplications of the form Ax or A^Ty to compute an
eps-feasible, eps-optimal solution. We show that FAL can be easily extended to
solve the basis pursuit denoising problem when there is a non-trivial level of
noise on the measurements. We report the results of numerical experiments
comparing FAL with the state-of-the-art algorithms for both noisy and noiseless
compressed sensing problems. A striking property of FAL that we observed in the
numerical experiments with randomly generated instances when there is no
measurement noise was that FAL always correctly identifies the support of the
target signal without any thresholding or post-processing, for moderately small
error tolerance values
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