Stochastic optimization

Stochastic optimization

Stochastic optimization (SO) methods are optimization algorithms which incorporate probabilistic (random) elements, either in the problem data (the objective function, the constraints, etc.), or in the algorithm itself (through random parameter values, random choices, etc.), or in both cite book
author = Spall, J. C.
title = Introduction to Stochastic Search and Optimization
year = 2003
publisher = Wiley
url = http://www.jhuapl.edu/ISSO] .The concept contrasts with the "deterministic" optimization methods, where the values of the objective function are assumed to be exact, and the computation is completely determined by the values sampled so far.

Methods for stochastic functions

Partly-random input data arise in such areas as real-time estimation and control, simulation-based optimization where Monte Carlo simulations are run as estimates of an actual system cite journal
author = Fu, M. C.
title = Optimization for Simulation: Theory vs. Practice
journal = INFORMS Journal on Computing
year = 2002
volume = 14
pages = 192–227
note = (with discussion by S. Andradóttir, P. Glynn, and J. P. Kelly)
doi = 10.1287/ijoc.14.3.192.113] , and problems where there is experimental (random) error in the measurements of the criterion. In such cases, knowledge that the function values are contaminated by random "noise" leads naturally to algorithms that use statistical inference tools to estimate the "true" values of the function and/or make statistically optimal decisions about the next steps. Methods of this class include
* stochastic approximation (SA), by Robbins and Monro (1951) cite journal
author = Robbins, H.
coauthors = Monro, S.
title = A Stochastic Approximation Method
journal = Annals of Mathematical Statistics
year = 1951
volume = 22
pages = 400–407
doi = 10.1214/aoms/1177729586]
** stochastic gradient descent
** finite-difference SA by Kiefer and Wolfowitz (1952) cite journal
author = J. Kiefer
authorlink = Jack Kiefer (mathematician)
coauthors = J. Wolfowitz
title = Stochastic Estimation of the Maximum of a Regression Function
journal = Annals of Mathematical Statistics
year = 1952
volume = 23
pages = 462–466
doi = 10.1214/aoms/1177729392]
** simultaneous perturbation SA by Spall (1992) cite journal
author = Spall, J. C.
title = Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation
journal = IEEE Transactions on Automatic Control
year = 1992
volume = 37
pages = 332–341
url = http://www.jhuapl.edu/SPSA
doi = 10.1109/9.119632]

Randomized search methods

On the other hand, even when the data is exact, it is sometimes beneficial to deliberately introduce randomness into the search process as a means of speeding convergence and making the algorithm less sensitive to modeling errors. Further, the injected randomness may provide the necessary impetus to move away from a local solution when searching for a global optimum. Indeed, this randomization principle is known to be a simple and effective way to obtain algorithms with "almost certain" good performance uniformly across all data sets, for all sorts of problems. Stochastic optimization method of this kind include:
* simulated annealing by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi (1983) cite journal
author = S. Kirkpatrick
coauthors = C. D. Gelatt; M. P. Vecchi
title = Optimization by Simulated Annealing
journal = Science
year = 1983
volume = Vol 220
number = 4598
pages = 671–680
url = http://citeseer.ist.psu.edu/kirkpatrick83optimization.html
doi = 10.1126/science.220.4598.671]
* evolutionary algorithms
** genetic algorithms by Goldberg (1989) cite book
author = Goldberg, D. E.
title = Genetic Algorithms in Search, Optimization, and Machine Learning
year = 1989
publisher = Addison-Wesley
url = http://www-illigal.ge.uiuc.edu]
* Cross-entropy method by Rubinstein and Kroese (2004) cite book
author = Rubinstein, R. Y
coauthors = Kroese, D.P
title = The Cross-Entropy Method
year = 2004
publisher = Springer-Verlag
url = http://www.cemethod.org]
* random search by Zhigljavsky (1991) cite book
author = Zhigljavsky, A. A.
title = Theory of Global Random Search
year = 1991
publisher = Kluwer Academic]
* Stochastic hill climbing

ee also

Machine Learning

References

*Michalewicz, Z. and Fogel, D. B. (2000), "How to Solve It: Modern Heuristics", Springer-Verlag, New York.

External links

oftware

* [http://www.aimms.com/aimms/context/mathematical_programming/stochastic_programming.html AIMMS]
* [http://www.optirisk-systems.com SPInE]
* [http://www.dash-optimization.com/home/products/products_sp.html XPRESS-SP]


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