1,076 research outputs found
Finiteness theorems in stochastic integer programming
We study Graver test sets for families of linear multi-stage stochastic
integer programs with varying number of scenarios. We show that these test sets
can be decomposed into finitely many ``building blocks'', independent of the
number of scenarios, and we give an effective procedure to compute these
building blocks. The paper includes an introduction to Nash-Williams' theory of
better-quasi-orderings, which is used to show termination of our algorithm. We
also apply this theory to finiteness results for Hilbert functions.Comment: 36 p
A polynomial oracle-time algorithm for convex integer minimization
In this paper we consider the solution of certain convex integer minimization
problems via greedy augmentation procedures. We show that a greedy augmentation
procedure that employs only directions from certain Graver bases needs only
polynomially many augmentation steps to solve the given problem. We extend
these results to convex -fold integer minimization problems and to convex
2-stage stochastic integer minimization problems. Finally, we present some
applications of convex -fold integer minimization problems for which our
approach provides polynomial time solution algorithms.Comment: 19 pages, 1 figur
A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs
In this paper we generalize N-fold integer programs and two-stage integer
programs with N scenarios to N-fold 4-block decomposable integer programs. We
show that for fixed blocks but variable N, these integer programs are
polynomial-time solvable for any linear objective. Moreover, we present a
polynomial-time computable optimality certificate for the case of fixed blocks,
variable N and any convex separable objective function. We conclude with two
sample applications, stochastic integer programs with second-order dominance
constraints and stochastic integer multi-commodity flows, which (for fixed
blocks) can be solved in polynomial time in the number of scenarios and
commodities and in the binary encoding length of the input data. In the proof
of our main theorem we combine several non-trivial constructions from the
theory of Graver bases. We are confident that our approach paves the way for
further extensions
Discounted continuous-time constrained Markov decision processes in Polish spaces
This paper is devoted to studying constrained continuous-time Markov decision
processes (MDPs) in the class of randomized policies depending on state
histories. The transition rates may be unbounded, the reward and costs are
admitted to be unbounded from above and from below, and the state and action
spaces are Polish spaces. The optimality criterion to be maximized is the
expected discounted rewards, and the constraints can be imposed on the expected
discounted costs. First, we give conditions for the nonexplosion of underlying
processes and the finiteness of the expected discounted rewards/costs. Second,
using a technique of occupation measures, we prove that the constrained
optimality of continuous-time MDPs can be transformed to an equivalent
(optimality) problem over a class of probability measures. Based on the
equivalent problem and a so-called -weak convergence of probability
measures developed in this paper, we show the existence of a constrained
optimal policy. Third, by providing a linear programming formulation of the
equivalent problem, we show the solvability of constrained optimal policies.
Finally, we use two computable examples to illustrate our main results.Comment: Published in at http://dx.doi.org/10.1214/10-AAP749 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
N-fold integer programming in cubic time
N-fold integer programming is a fundamental problem with a variety of natural
applications in operations research and statistics. Moreover, it is universal
and provides a new, variable-dimension, parametrization of all of integer
programming. The fastest algorithm for -fold integer programming predating
the present article runs in time with the binary length of
the numerical part of the input and the so-called Graver complexity of
the bimatrix defining the system. In this article we provide a drastic
improvement and establish an algorithm which runs in time having
cubic dependency on regardless of the bimatrix . Our algorithm can be
extended to separable convex piecewise affine objectives as well, and also to
systems defined by bimatrices with variable entries. Moreover, it can be used
to define a hierarchy of approximations for any integer programming problem
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