202 research outputs found
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
Some Trace Inequalities for Operators in Hilbert Spaces
Some new trace inequalities for operators in Hilbert spaces are provided. The
superadditivity and monotonicity of some associated functionals are
investigated and applications for power series of such operators are given.
Some trace inequalities for matrices are also derived. Examples for the
operator exponential and other similar functions are presented as well
Composable computation in discrete chemical reaction networks
We study the composability of discrete chemical reaction networks (CRNs) that
stably compute (i.e., with probability 0 of error) integer-valued functions
. We consider output-oblivious CRNs in which the
output species is never a reactant (input) to any reaction. The class of
output-oblivious CRNs is fundamental, appearing in earlier studies of CRN
computation, because it is precisely the class of CRNs that can be composed by
simply renaming the output of the upstream CRN to match the input of the
downstream CRN.
Our main theorem precisely characterizes the functions stably computable
by output-oblivious CRNs with an initial leader. The key necessary condition is
that for sufficiently large inputs, is the minimum of a finite number of
nondecreasing quilt-affine functions. (An affine function is linear with a
constant offset; a quilt-affine function is linear with a periodic offset)
Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)
We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is defined as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution differs from the Myerson-kind value defined in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game. For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it by using its Möbius inversion.Core;polytope;building set;nested set complex;Möbius inversion;permutations;normal fan;average tree solution;Myerson value
Multivariate risk measures : a constructive approach based on selections
Since risky positions in multivariate portfolios can be offset by various choices of
capital requirements that depend on the exchange rules and related transaction costs, it
is natural to assume that the risk measures of random vectors are set-valued.
Furthermore, it is reasonable to include the exchange rules in the argument of the risk
and so consider risk measures of set-valued portfolios. This situation includes the
classical Kabanov's transaction costs model, where the set-valued portfolio is given by
the sum of a random vector and an exchange cone, but also a number of further cases of
additional liquidity constraints.
The definition of the selection risk measure is based on calling a set-valued portfolio
acceptable if it possesses a selection with all individually acceptable marginals. The
obtained risk measure is coherent (or convex), law invariant and has values being upper
convex closed sets. We describe the dual representation of the selection risk measure
and suggest efficient ways of approximating it from below and from above. In case of
Kabanov's exchange cone model, it is shown how the selection risk measure relates to
the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010),
and Hamel et al. (2013)Supported by the Spanish Ministry of Science and Innovation Grants No. MTM20II—22993 and ECO20ll-25706. Supported by the Chair of Excellence Programme of the Universidad Carlos III de Madrid and Banco
Santander and the Swiss National Foundation Grant No. 200021-13752
A Discrete Choquet Integral for Ordered Systems
A model for a Choquet integral for arbitrary finite set systems is presented.
The model includes in particular the classical model on the system of all
subsets of a finite set. The general model associates canonical non-negative
and positively homogeneous superadditive functionals with generalized belief
functions relative to an ordered system, which are then extended to arbitrary
valuations on the set system. It is shown that the general Choquet integral can
be computed by a simple Monge-type algorithm for so-called intersection
systems, which include as a special case weakly union-closed families.
Generalizing Lov\'asz' classical characterization, we give a characterization
of the superadditivity of the Choquet integral relative to a capacity on a
union-closed system in terms of an appropriate model of supermodularity of such
capacities
Limiting shape for directed percolation models
We consider directed first-passage and last-passage percolation on the
nonnegative lattice Z_+^d, d\geq2, with i.i.d. weights at the vertices. Under
certain moment conditions on the common distribution of the weights, the limits
g(x)=lim_{n\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for
x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in
Z_+^d. We show that this shape function g is continuous on R_+^d, in particular
at the boundaries. In two dimensions, we give more precise asymptotics for the
behavior of g near the boundaries; these asymptotics depend on the common
weight distribution only through its mean and variance. In addition we discuss
growth models which are naturally associated to the percolation processes,
giving a shape theorem and illustrating various possible types of behavior with
output from simulations.Comment: Published at http://dx.doi.org/10.1214/009117904000000838 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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