In recent years, there has been a growing interest in developing statistical learning methods to provide approximate solutions to "difficult" control problems. In particular, randomized algorithms have becom...
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In recent years, there has been a growing interest in developing statistical learning methods to provide approximate solutions to "difficult" control problems. In particular, randomized algorithms have become a very popular tool used for stability and performance analysis as well as for design of control systems. However, as randomized algorithms provide an efficient solution procedure to the "intractable" problems, stochastic methods bring closer to understanding the properties of the real systems. The topic of this paper is the use of stochastic methods in order to solve the problem of control robustness: the case of parametric stochastic uncertainty is considered. Necessary concepts regarding stochastic control theory and stochastic differential equations are introduced. Then a convergence analysis is provided by means of the Chernoff bounds, which guarantees robustness in mean and in probability. As an illustration, the robustness of control performances of example control systems is computed. (C) 2009 Elsevier Ltd. All rights reserved.
In this note, we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities (LMIs) of the form F(x, Delta) less than or equal to 0, where 0 is the optimization vari...
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In this note, we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities (LMIs) of the form F(x, Delta) less than or equal to 0, where 0 is the optimization variable and Delta is the uncertainty, which belongs to a given set Delta. The proposed algorithms are based on uncertainty randomization: the first algorithm finds a robust solution in a finite number of iterations with probability one, if a strong feasibility condition holds. In case no robust solution exists, the second algorithm computes an approximate solution which minimizes the expected value of a suitably selected feasibility indicator function. The theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices.
Recently, the authors have described two stochastic algorithms, based on the controlled random search, for the global optimization. This paper deals with the use of these algorithms in estimating the parameters of non...
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Recently, the authors have described two stochastic algorithms, based on the controlled random search, for the global optimization. This paper deals with the use of these algorithms in estimating the parameters of nonlinear regression models. Several criteria like residual sum of squares, sum of absolute deviations and sum of trimmed squares are chosen. The algorithms are experimentally tested on a set of the well-known tasks chosen in such way that most classical techniques based on objective function derivatives fail while treating them. The basic features of the algorithms (rate of convergence and reliability) as well as their applicability to nonlinear regression models are discussed in more detail. (C) 2000 Elsevier Science B.V. All rights reserved.
This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. The algorithm computes equilibrium policy and value functions, and generates a transition kernel for the (stochastic) evo...
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This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. The algorithm computes equilibrium policy and value functions, and generates a transition kernel for the (stochastic) evolution of the state of the system, It has two features that together imply that it need not be subject to the curse of dimensionality. First, the integral that determines continuation values is never calculated;rather it is approximated by a simple average of returns from past outcomes of the algorithm, an approximation whose computational burden is not tied to the dimension of the state space. Second, iterations of the algorithm update value and policy functions at a single (rather than at all possible) points in the state space. Random draws from a distribution set by the updated policies determine the location of the next iteration's updates. This selection only repeatedly hits the recurrent class of points, a subset whose cardinality is not directly tied to that of the state space. Numerical results for industrial organization problems show that our algorithm can increase speed and decrease memory requirements by several orders of magnitude.
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.
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.
Optimization with stochastic algorithms has become a relevant approach, specially, in problems with complex search spaces. Due to the stochastic nature of these algorithms, the assessment and comparison is not straigh...
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ISBN:
(纸本)9781450305570
Optimization with stochastic algorithms has become a relevant approach, specially, in problems with complex search spaces. Due to the stochastic nature of these algorithms, the assessment and comparison is not straightforward. Several performance measures have been proposed to overcome this difficulty. In this work, the use of performance profiles and an analysis integrating a trade-off between accuracy and precision are carried out for the comparison of two stochastic algorithms. Traditionally, performance profiles are used to compare deterministic algorithms. This methodology is applied in the comparison of two stochastic algorithms - genetic algorithms and simulated annealing. The results highlight the advantages and drawbacks of the proposed assessment. Track Name: Genetic algorithms
Modern machine learning has drawn a lot of attention from many researchers, showing promising directions where some of large-scale computational challenges become tractable, especially in material science and chemistr...
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Modern machine learning has drawn a lot of attention from many researchers, showing promising directions where some of large-scale computational challenges become tractable, especially in material science and chemistry. In this dissertation, we explore several problems in computational chemistry and machine learning in the perspective of mathematics. We develop stochastic algorithms supplemented with rigorous analysis, especially for self-consistent computations in electronic structure theory. We also present a local convergence analysis of the stochastic gradient descent algorithms for non-convex models in machine learning. Lastly, we demonstrate the effectiveness of the diffusion map to ab-initio molecular dynamics data for identifying reaction collective variables and computing committor ***-consistent calculations of electronic structure have recently become routine computations for many applications such as the analysis of material properties and the simulation of molecular dynamics [33, 70, 87, 88, 90, 103, 117, 126]. The computational cost of standard self-consistent calculations increases cubically with respect to system size (e.g. density functional theory (DFT) [4, 55, 68, 70, 120] or self-consistent charge density functional tight-binding (SCC-DFTB) [33] ). The quantities of interest in DFT and SCC-DFTB are determined by the electron charges, which can be formulated as a diagonal of a matrix function. Many numerical methods have been proposed to speed up self-consistent calculations [3, 14–16, 18, 19, 35, 51, 59, 69, 81, 105, 132, 135, 136]. However, standard self-consistent calculations for computing electron charges require the eigenvalues and eigenvectors of such a matrix function, which makes those calculations intractable for large electronic systems. Chapter 1 of this dissertation presents a stochastic framework for the self-consistent calculations, which formulates the electron charges as a conditional expectation using the diagonal esti
A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains seve...
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A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed on a benchmark test case, where certain advantages of the new class of algorithms are demonstrated.
We study the almost sure asymptotic behaviour of decreasing stepsized stochastic algorithms used for the search of zeros of a function. We prove a law of the iterated logarithm, which gives the almost sure convergence...
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We study the almost sure asymptotic behaviour of decreasing stepsized stochastic algorithms used for the search of zeros of a function. We prove a law of the iterated logarithm, which gives the almost sure convergence rate of the algorithm, and we establish a quadratic strong law of large numbers. (C) 1998 Published by Elsevier Science B.V. All rights reserved.
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