7,187 research outputs found
On the expected diameter, width, and complexity of a stochastic convex-hull
We investigate several computational problems related to the stochastic
convex hull (SCH). Given a stochastic dataset consisting of points in
each of which has an existence probability, a SCH refers to the
convex hull of a realization of the dataset, i.e., a random sample including
each point with its existence probability. We are interested in computing
certain expected statistics of a SCH, including diameter, width, and
combinatorial complexity. For diameter, we establish the first deterministic
1.633-approximation algorithm with a time complexity polynomial in both and
. For width, two approximation algorithms are provided: a deterministic
-approximation running in time, and a fully
polynomial-time randomized approximation scheme (FPRAS). For combinatorial
complexity, we propose an exact -time algorithm. Our solutions exploit
many geometric insights in Euclidean space, some of which might be of
independent interest
Stochastic Combinatorial Optimization via Poisson Approximation
We study several stochastic combinatorial problems, including the expected
utility maximization problem, the stochastic knapsack problem and the
stochastic bin packing problem. A common technical challenge in these problems
is to optimize some function of the sum of a set of random variables. The
difficulty is mainly due to the fact that the probability distribution of the
sum is the convolution of a set of distributions, which is not an easy
objective function to work with. To tackle this difficulty, we introduce the
Poisson approximation technique. The technique is based on the Poisson
approximation theorem discovered by Le Cam, which enables us to approximate the
distribution of the sum of a set of random variables using a compound Poisson
distribution.
We first study the expected utility maximization problem introduced recently
[Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we
obtain an additive PTAS if there is a multidimensional PTAS for the
multi-objective version of the problem, strictly generalizing the previous
result.
For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and
Tardos, STOC97]), we show there is a polynomial time algorithm which uses at
most the optimal number of bins, if we relax the size of each bin and the
overflow probability by eps.
For stochastic knapsack, we show a 1+eps-approximation using eps extra
capacity, even when the size and reward of each item may be correlated and
cancelations of items are allowed. This generalizes the previous work [Balghat,
Goel and Khanna, SODA11] for the case without correlation and cancelation. Our
algorithm is also simpler. We also present a factor 2+eps approximation
algorithm for stochastic knapsack with cancelations. the current known
approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of
the 45th ACM Symposium on the Theory of Computing (STOC13
Improved decision support for engine-in-the-loop experimental design optimization
Experimental optimization with hardware in the loop is a common procedure in engineering and has been the subject of intense development, particularly when it is applied to relatively complex combinatorial systems that are not completely understood, or where accurate modelling is not possible owing to the dimensions of the search space. A common source of difficulty arises because of the level of noise associated with experimental measurements, a combination of limited instrument precision, and extraneous factors. When a series of experiments is conducted to search for a combination of input parameters that results in a minimum or maximum response, under the imposition of noise, the underlying shape of the function being optimized can become very difficult to discern or even lost. A common methodology to support experimental search for optimal or suboptimal values is to use one of the many gradient descent methods. However, even sophisticated and proven methodologies, such as simulated annealing, can be significantly challenged in the presence of noise, since approximating the gradient at any point becomes highly unreliable. Often, experiments are accepted as a result of random noise which should be rejected, and vice versa. This is also true for other sampling techniques, including tabu and evolutionary algorithms.
After the general introduction, this paper is divided into two main sections (sections 2 and 3), which are followed by the conclusion. Section 2 introduces a decision support methodology based upon response surfaces, which supplements experimental management based on a variable neighbourhood search and is shown to be highly effective in directing experiments in the presence of a significant signal-to-noise ratio and complex combinatorial functions. The methodology is developed on a three-dimensional surface with multiple local minima, a large basin of attraction, and a high signal-to-noise ratio.
In section 2, the methodology is applied to an automotive combinatorial search in the laboratory, on a real-time engine-in-the-loop application. In this application, it is desired to find the maximum power output of an experimental single-cylinder spark ignition engine operating under a quasi-constant-volume operating regime. Under this regime, the piston is slowed at top dead centre to achieve combustion in close to constant volume conditions.
As part of the further development of the engine to incorporate a linear generator to investigate free-piston operation, it is necessary to perform a series of experiments with combinatorial parameters. The objective is to identify the maximum power point in the least number of experiments in order to minimize costs. This test programme provides peak power data in order to achieve optimal electrical machine design.
The decision support methodology is combined with standard optimization and search methods – namely gradient descent and simulated annealing – in order to study the reductions possible in experimental iterations. It is shown that the decision support methodology significantly reduces the number of experiments necessary to find the maximum power solution and thus offers a potentially significant cost saving to hardware-in-the-loop experi- mentation
Stochastic Vehicle Routing with Recourse
We study the classic Vehicle Routing Problem in the setting of stochastic
optimization with recourse. StochVRP is a two-stage optimization problem, where
demand is satisfied using two routes: fixed and recourse. The fixed route is
computed using only a demand distribution. Then after observing the demand
instantiations, a recourse route is computed -- but costs here become more
expensive by a factor lambda.
We present an O(log^2 n log(n lambda))-approximation algorithm for this
stochastic routing problem, under arbitrary distributions. The main idea in
this result is relating StochVRP to a special case of submodular orienteering,
called knapsack rank-function orienteering. We also give a better approximation
ratio for knapsack rank-function orienteering than what follows from prior
work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of
approximation for StochVRP, even on star-like metrics on which our algorithm
achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of
Theorem 1.
Deterministic Annealing and Nonlinear Assignment
For combinatorial optimization problems that can be formulated as Ising or
Potts spin systems, the Mean Field (MF) approximation yields a versatile and
simple ANN heuristic, Deterministic Annealing. For assignment problems the
situation is more complex -- the natural analog of the MF approximation lacks
the simplicity present in the Potts and Ising cases. In this article the
difficulties associated with this issue are investigated, and the options for
solving them discussed. Improvements to existing Potts-based MF-inspired
heuristics are suggested, and the possibilities for defining a proper
variational approach are scrutinized.Comment: 15 pages, 3 figure
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
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