In this article, we consider a Linear Programming (LP) problem with unknown objective function. We introduce a class of stochastic algorithms to estimate an optimal solution of the LP problem. The almost sure converge...
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In this article, we consider a Linear Programming (LP) problem with unknown objective function. We introduce a class of stochastic algorithms to estimate an optimal solution of the LP problem. The almost sure convergence and the speed of convergence of these algorithms are analyzed. We also prove a central limit theorem for the estimation errors of the algorithms.
A device has two arms with unknown deterministic payoffs and the aim is to asymptotically identify the best one without spending too much time on the other. The Narendra algorithm offers a stochastic procedure to this...
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A device has two arms with unknown deterministic payoffs and the aim is to asymptotically identify the best one without spending too much time on the other. The Narendra algorithm offers a stochastic procedure to this end. We show under weak ergodic assumptions on these deterministic payoffs that the procedure eventually chooses the best arm (i.e., with greatest Cesaro limit) with probability one for appropriate step sequences of the algorithm. In the case of i.i.d. payoffs, this implies a "quenched" version of the "annealed" result of Lamberton, Pages and Tarres [Ann. Appl. Probab. 14 (2004) 1424-1454] by the law of iterated logarithm, thus generalizing it. More precisely, if (eta(l),i)(i is an element of N) is an element of {0, 1}(N), l is an element of {A, B}, are the deterministic reward sequences we would get if we played at time i, we obtain infallibility with the same assumption on nonincreasing step sequences on the payoffs as in Lamberton, Pages and Tarres [Ann. Appl. Probab. 14 (2004) 1424-1454], replacing the i.i.d. assumption by the hypothesis that the empirical averages Sigma(n)(i=1) eta(A,i)/n and Sigma(n)(i=1) eta(B,i)/n converge, as n tends to infinity, respectively, to theta(A) and theta(B), with rate at least 1/(log n)(1+epsilon), for some epsilon > 0. We also show a fallibility result, that is, convergence with positive probability to the choice of the wrong arm, which implies the corresponding result of Larnberton, Pages and Tarres [Ann. Appl. Probab. 14 (2004) 1424-1454] in the i.i.d. case.
This work investigates the problem of construction of designs for estimation and discrimination between competing linear models. In our framework, the unknown signal is observed with the addition of a noise and only a...
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This work investigates the problem of construction of designs for estimation and discrimination between competing linear models. In our framework, the unknown signal is observed with the addition of a noise and only a few evaluations of the noisy signal are available. The model selection is performed in a multi-resolution setting. In this setting, the locations of discrete sequential D and A designs are precisely constraint in a small number of explicit points. Hence, an efficient stochastic algorithm can be constructed that alternately improves the design and the model. Several numerical experiments illustrate the efficiency of our method for regression. One can also use this algorithm as a preliminary step to build response surfaces for sensitivity analysis.
This paper studies the stochastic behavior of the LMS algorithm for a system identification framework when the input signal is a non-stationary white Gaussian process. The unknown system is modeled by the standard ran...
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ISBN:
(纸本)9781457705700
This paper studies the stochastic behavior of the LMS algorithm for a system identification framework when the input signal is a non-stationary white Gaussian process. The unknown system is modeled by the standard random walk model. An approximate theory is developed which is based upon the instantaneous average power in the adaptive filter taps. The stability of the algorithm is investigated using this model. Monte Carlo simulations of the algorithm provides strong support for the theoretical approximation.
Normalized forms of adaptive algorithms are usually sought in order to obtain convergence properties independent of the input signal power. Such is the case of the well-known Normalized LMS (NLMS) algorithm. The Least...
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Normalized forms of adaptive algorithms are usually sought in order to obtain convergence properties independent of the input signal power. Such is the case of the well-known Normalized LMS (NLMS) algorithm. The Least-Mean Fourth (LMF) adaptive algorithm has been shown to outperform LMS in different situations. However, the LMF stability is dependent on both the signal power and on the adaptive weights initialization. This paper studies the behavior of two normalized forms of the LMF algorithm for Gaussian inputs. Contrary to what could be expected, the mean-square stability of both normalized algorithms is shown to be dependent upon the input signal power. Thus, the usefulness of the NLMF algorithm is open to question. (C) 2011 Elsevier Inc. All rights reserved.
A geophysical interpretative method is proposed to depth, amplitude coefficient (effective magnetization intensity), and index parameter (effective magnetization inclination) determination of a buried structure from m...
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A geophysical interpretative method is proposed to depth, amplitude coefficient (effective magnetization intensity), and index parameter (effective magnetization inclination) determination of a buried structure from magnetic field data anomaly due to a fault, a thin dike or a sphere-like structure. The method is based on the nonlinearly constrained mathematical modelling and also on the stochastic optimization approaches. The proposed interpretative method was first tested on a theoretical synthetic model with different random errors, where a very close agreement was obtained between the assumed and the evaluated parameters. The validity of this method was also tested on practical field data taken from United States, Australia, India and Brazil, where available magnetic data existed and were previously analyzed by different interpretative methods. The agreement between the results obtained by our developed method and those obtained by the other geophysical methods is good.
We consider a distributed multi-agent network system where the goal is to minimize the sum of convex functions, each of which is known (with stochastic errors) to a specific network agent. We are interested in asynchr...
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ISBN:
(纸本)9781424470143
We consider a distributed multi-agent network system where the goal is to minimize the sum of convex functions, each of which is known (with stochastic errors) to a specific network agent. We are interested in asynchronous algorithms for solving the problem over a connected network where the communications among the agent are random. At each time, a random set of agents communicate and update their information. When updating, an agent uses the (sub) gradient of its individual objective function and its own stepsize value. The algorithm is completely asynchronous as it neither requires the coordination of agent actions nor the coordination of the stepsize values. We investigate the asymptotic error bounds of the algorithm with a constant stepsize for strongly convex and just convex functions. Our error bounds capture the effects of agent stepsize choices and the structure of the agent connectivity graph. The error bound scales at best as m in the number m of agents when the agent objective functions are strongly convex.
This research concerns permutation Flow Shop scheduling (in this case, a schedule is a total order on the jobs). In Flow-Shop problems, stochastic algorithms have been largely used to minimize the makespan or the tota...
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This research concerns permutation Flow Shop scheduling (in this case, a schedule is a total order on the jobs). In Flow-Shop problems, stochastic algorithms have been largely used to minimize the makespan or the total completion time. Usually, initial solutions are computed from heuristics. This paper shows that for stochastic algorithms, better results can be obtained by first looking at the worst solution (maximizing the criteria) then reversing the sequence and finally using this reverse solution as an initial solution. The quality of the results is improved for the makespan and the total completion time. The repeatability of the stochastic algorithm is also largely improved. It therefore appears that looking for the worst solution can be efficient in the search of the best. (c) 2005 Elsevier B.V. All rights reserved.
A stochastic process that describes a payoff-based learning procedure and the associated adaptive behavior of players in a repeated game is considered. The process is shown to converge almost surely towards a stationa...
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A stochastic process that describes a payoff-based learning procedure and the associated adaptive behavior of players in a repeated game is considered. The process is shown to converge almost surely towards a stationary state which is characterized as an equilibrium for a related game. The analysis is based on techniques borrowed from the theory of stochastic algorithms and proceeds by studying an associated continuous dynamical system which represents the evolution of the players' evaluations. An application to the case of finitely many users in a congested traffic network with parallel links is considered. Alternative descriptions for the dynamics and the corresponding rest points are discussed, including a Lagrangian representation. (C) 2008 Elsevier Inc. All rights reserved.
This research concerns permutation Flow Shop scheduling (in this case, a schedule is a total order on the jobs). In Flow-Shop problems, stochastic algorithms have been largely used to minimize the makespan or the tota...
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This research concerns permutation Flow Shop scheduling (in this case, a schedule is a total order on the jobs). In Flow-Shop problems, stochastic algorithms have been largely used to minimize the makespan or the total completion time. Usually, initial solutions are computed from heuristics. This paper shows that for stochastic algorithms, better results can be obtained by first looking at the worst solution (maximizing the criteria) then reversing the sequence and finally using this reverse solution as an initial solution. The quality of the results is improved for the makespan and the total completion time. The repeatability of the stochastic algorithm is also largely improved. It therefore appears that looking for the worst solution can be efficient in the search of the best. (c) 2005 Elsevier B.V. All rights reserved.
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