991 research outputs found
Split rank of triangle and quadrilateral inequalities
A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen et al. [2] and Cornu´ejols and Margot [13] showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Through a result by Cook et al. [12], it is known that one particular class of facet- defining triangle inequality does not have a finite split rank. In this paper, we show that all other facet-defining triangle and quadrilateral inequalities have finite split rank. The proof is constructive and given a facet-defining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.mixed integer programs, split rank, group relaxations
Integral Farkas type lemmas for systems with equalities and inequalities
A central result in the theory of integer optimization states that a system of linear diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, y T A integral such that y T b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, Cx d}. We show that a certificate of integral infeasibility is a linear system with rank(C) variables containing no integral point.
Min Max Generalization for Two-stage Deterministic Batch Mode Reinforcement Learning: Relaxation Schemes
We study the minmax optimization problem introduced in [22] for computing
policies for batch mode reinforcement learning in a deterministic setting.
First, we show that this problem is NP-hard. In the two-stage case, we provide
two relaxation schemes. The first relaxation scheme works by dropping some
constraints in order to obtain a problem that is solvable in polynomial time.
The second relaxation scheme, based on a Lagrangian relaxation where all
constraints are dualized, leads to a conic quadratic programming problem. We
also theoretically prove and empirically illustrate that both relaxation
schemes provide better results than those given in [22]
Weighted Sum Rate Maximization for Downlink OFDMA with Subcarrier-pair based Opportunistic DF Relaying
This paper addresses a weighted sum rate (WSR) maximization problem for
downlink OFDMA aided by a decode-and-forward (DF) relay under a total power
constraint. A novel subcarrier-pair based opportunistic DF relaying protocol is
proposed. Specifically, user message bits are transmitted in two time slots. A
subcarrier in the first slot can be paired with a subcarrier in the second slot
for the DF relay-aided transmission to a user. In particular, the source and
the relay can transmit simultaneously to implement beamforming at the
subcarrier in the second slot. Each unpaired subcarrier in either the first or
second slot is used for the source's direct transmission to a user. A benchmark
protocol, same as the proposed one except that the transmit beamforming is not
used for the relay-aided transmission, is also considered. For each protocol, a
polynomial-complexity algorithm is developed to find at least an approximately
optimum resource allocation (RA), by using continuous relaxation, the dual
method, and Hungarian algorithm. Instrumental to the algorithm design is an
elegant definition of optimization variables, motivated by the idea of
regarding the unpaired subcarriers as virtual subcarrier pairs in the direct
transmission mode. The effectiveness of the RA algorithm and the impact of
relay position and total power on the protocols' performance are illustrated by
numerical experiments. The proposed protocol always leads to a maximum WSR
equal to or greater than that for the benchmark one, and the performance gain
of using the proposed one is significant especially when the relay is in close
proximity to the source and the total power is low. Theoretical analysis is
presented to interpret these observations.Comment: 8 figures, accepted and to be published in IEEE Transactions on
Signal Processing. arXiv admin note: text overlap with arXiv:1301.293
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