449 research outputs found
A Newton-bracketing method for a simple conic optimization problem
For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs),
we propose a Newton-bracketing method to improve the performance of the
bisection-projection method implemented in BBCPOP [to appear in ACM Tran.
Softw., 2019]. The relaxation problem is converted into the problem of finding
the largest zero of a continuously differentiable (except at )
convex function such that if
and otherwise. In theory, the method generates lower
and upper bounds of both converging to . Their convergence is
quadratic if the right derivative of at is positive. Accurate
computation of is necessary for the robustness of the method, but it is
difficult to achieve in practice. As an alternative, we present a
secant-bracketing method. We demonstrate that the method improves the quality
of the lower bounds obtained by BBCPOP and SDPNAL+ for binary QOP instances
from BIQMAC. Moreover, new lower bounds for the unknown optimal values of large
scale QAP instances from QAPLIB are reported.Comment: 19 pages, 2 figure
Global Approaches for Facility Layout and VLSI Floorplanning
This paper summarizes recent advances in the global solution of several relevant facility layout problems
Global Approaches for Facility Layout and VLSI Floorplanning
This paper summarizes recent advances in the global solution of several relevant facility layout problems
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)
AMS classification: 90C22, 20Cxx, 70-08
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