393 research outputs found
Exact Solution Methods for the -item Quadratic Knapsack Problem
The purpose of this paper is to solve the 0-1 -item quadratic knapsack
problem , a problem of maximizing a quadratic function subject to two
linear constraints. We propose an exact method based on semidefinite
optimization. The semidefinite relaxation used in our approach includes simple
rank one constraints, which can be handled efficiently by interior point
methods. Furthermore, we strengthen the relaxation by polyhedral constraints
and obtain approximate solutions to this semidefinite problem by applying a
bundle method. We review other exact solution methods and compare all these
approaches by experimenting with instances of various sizes and densities.Comment: 12 page
Stochastic Budget Optimization in Internet Advertising
Internet advertising is a sophisticated game in which the many advertisers
"play" to optimize their return on investment. There are many "targets" for the
advertisements, and each "target" has a collection of games with a potentially
different set of players involved. In this paper, we study the problem of how
advertisers allocate their budget across these "targets". In particular, we
focus on formulating their best response strategy as an optimization problem.
Advertisers have a set of keywords ("targets") and some stochastic information
about the future, namely a probability distribution over scenarios of cost vs
click combinations. This summarizes the potential states of the world assuming
that the strategies of other players are fixed. Then, the best response can be
abstracted as stochastic budget optimization problems to figure out how to
spread a given budget across these keywords to maximize the expected number of
clicks.
We present the first known non-trivial poly-logarithmic approximation for
these problems as well as the first known hardness results of getting better
than logarithmic approximation ratios in the various parameters involved. We
also identify several special cases of these problems of practical interest,
such as with fixed number of scenarios or with polynomial-sized parameters
related to cost, which are solvable either in polynomial time or with improved
approximation ratios. Stochastic budget optimization with scenarios has
sophisticated technical structure. Our approximation and hardness results come
from relating these problems to a special type of (0/1, bipartite) quadratic
programs inherent in them. Our research answers some open problems raised by
the authors in (Stochastic Models for Budget Optimization in Search-Based
Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
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