2,617 research outputs found
An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging
This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions
A Duality Theory with Zero Duality Gap for Nonlinear Programming
Duality is an important notion for constrained optimization which provides a theoretical foundation for a number of constraint decomposition schemes such as separable programming and for deriving lower bounds in space decomposition algorithms such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex optimization problems, especially discrete and mixed-integer problems where the feasible sets are nonconvex. In this paper, we propose a novel extended duality theory for nonlinear optimization that overcomes some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed-integer spaces under mild conditions
Joint Scheduling and Resource Allocation in the OFDMA Downlink: Utility Maximization under Imperfect Channel-State Information
We consider the problem of simultaneous user-scheduling, power-allocation,
and rate-selection in an OFDMA downlink, with the goal of maximizing expected
sum-utility under a sum-power constraint. In doing so, we consider a family of
generic goodput-based utilities that facilitate, e.g., throughput-based
pricing, quality-of-service enforcement, and/or the treatment of practical
modulation-and-coding schemes (MCS). Since perfect knowledge of channel state
information (CSI) may be difficult to maintain at the base-station, especially
when the number of users and/or subchannels is large, we consider scheduling
and resource allocation under imperfect CSI, where the channel state is
described by a generic probability distribution. First, we consider the
"continuous" case where multiple users and/or code rates can time-share a
single OFDMA subchannel and time slot. This yields a non-convex optimization
problem that we convert into a convex optimization problem and solve exactly
using a dual optimization approach. Second, we consider the "discrete" case
where only a single user and code rate is allowed per OFDMA subchannel per time
slot. For the mixed-integer optimization problem that arises, we discuss the
connections it has with the continuous case and show that it can solved exactly
in some situations. For the other situations, we present a bound on the
optimality gap. For both cases, we provide algorithmic implementations of the
obtained solution. Finally, we study, numerically, the performance of the
proposed algorithms under various degrees of CSI uncertainty, utilities, and
OFDMA system configurations. In addition, we demonstrate advantages relative to
existing state-of-the-art algorithms
Strong duality in conic linear programming: facial reduction and extended duals
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual
of Ramana provide a strong dual for the conic linear program in the absence of any constraint qualification. The facial
reduction algorithm solves a sequence of auxiliary optimization problems to
obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite
program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz
showed that these approaches are closely related; in particular, they proved
the correctness of Ramana's dual using certificates from a facial reduction
algorithm.
Here we give a clear and self-contained exposition of facial reduction, of
extended duals, and generalize Ramana's dual:
-- we state a simple facial reduction algorithm and prove its correctness;
and
-- building on this algorithm we construct a family of extended duals when
is a {\em nice} cone. This class of cones includes the semidefinite cone
and other important cones.Comment: A previous version of this paper appeared as "A simple derivation of
a facial reduction algorithm and extended dual systems", technical report,
Columbia University, 2000, available from
http://www.unc.edu/~pataki/papers/fr.pdf Jonfest, a conference in honor of
Jonathan Borwein's 60th birthday, 201
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