8,076 research outputs found
Statistical mechanics of the multi-constraint continuous knapsack problem
We apply the replica analysis established by Gardner to the multi-constraint
continuous knapsack problem,which is one of the linear programming problems and
a most fundamental problem in the field of operations research (OR). For a
large problem size, we analyse the space of solution and its volume, and
estimate the optimal number of items to go into the knapsack as a function of
the number of constraints. We study the stability of the replica symmetric (RS)
solution and find that the RS calculation cannot estimate the optimal number of
items in knapsack correctly if many constraints are required.In order to obtain
a consistent solution in the RS region,we try the zero entropy approximation
for this continuous solution space and get a stable solution within the RS
ansatz.On the other hand, in replica symmetry breaking (RSB) region, the one
step RSB solution is found by Parisi's scheme. It turns out that this problem
is closely related to the problem of optimal storage capacity and of
generalization by maximum stability rule of a spherical perceptron.Comment: Latex 13 pages using IOP style file, 5 figure
Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies
Patriksson (2008) provided a then up-to-date survey on the
continuous,separable, differentiable and convex resource allocation problem
with a single resource constraint. Since the publication of that paper the
interest in the problem has grown: several new applications have arisen where
the problem at hand constitutes a subproblem, and several new algorithms have
been developed for its efficient solution. This paper therefore serves three
purposes. First, it provides an up-to-date extension of the survey of the
literature of the field, complementing the survey in Patriksson (2008) with
more then 20 books and articles. Second, it contributes improvements of some of
these algorithms, in particular with an improvement of the pegging (that is,
variable fixing) process in the relaxation algorithm, and an improved means to
evaluate subsolutions. Third, it numerically evaluates several relaxation
(primal) and breakpoint (dual) algorithms, incorporating a variety of pegging
strategies, as well as a quasi-Newton method. Our conclusion is that our
modification of the relaxation algorithm performs the best. At least for
problem sizes up to 30 million variables the practical time complexity for the
breakpoint and relaxation algorithms is linear
A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger
We present an adaptive version of the Multi-Index Monte Carlo method,
introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with
coefficients that are random fields. A classical technique for sampling from
these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm
is based on the adaptive algorithm used in sparse grid cubature as introduced
by Gerstner and Griebel (2003), and automatically chooses the number of terms
needed in this expansion, as well as the required spatial discretizations of
the PDE model. We apply the method to a simplified model of a heat exchanger
with random insulator material, where the stochastic characteristics are
modeled as a lognormal random field, and we show consistent computational
savings
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