454 research outputs found
SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression
This paper deals with the problem of finding the globally optimal subset of h
elements from a larger set of n elements in d space dimensions so as to
minimize a quadratic criterion, with an special emphasis on applications to
computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The
computation of the LTSE is a challenging subset selection problem involving a
nonlinear program with continuous and binary variables, linked in a highly
nonlinear fashion. The selection of a globally optimal subset using the branch
and bound (BB) algorithm is limited to problems in very low dimension,
tipically d<5, as the complexity of the problem increases exponentially with d.
We introduce a bold pruning strategy in the BB algorithm that results in a
significant reduction in computing time, at the price of a negligeable accuracy
lost. The novelty of our algorithm is that the bounds at nodes of the BB tree
come from pseudo-convexifications derived using a linearization technique with
approximate bounds for the nonlinear terms. The approximate bounds are computed
solving an auxiliary semidefinite optimization problem. We show through a
computational study that our algorithm performs well in a wide set of the most
difficult instances of the LTSE problem.Comment: 12 pages, 3 figures, 2 table
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
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
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