Stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective functions or random constraints. Stochastic optimization methods also include methods with random iterates. Some stochastic optimization methods use random iterates to solve stochastic problems, combining both meanings of stochastic optimization.[1] Stochastic optimization methods generalize deterministic methods for deterministic problems.
Methods for stochastic functions
Partly random input data arise in such areas as realtime estimation and control, simulationbased optimization where Monte Carlo simulations are run as estimates of an actual system,[2] [3] 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)[4]
 stochastic gradient descent
 finitedifference SA by Kiefer and Wolfowitz (1952)[5]
 simultaneous perturbation SA by Spall (1992)[6]
 scenario optimization
Randomized search methods
On the other hand, even when the data set consists of precise measurements, some methods introduce randomness into the searchprocess to accelerate progress.[7] Such randomness can also make the method less sensitive to modeling errors. Further, the injected randomness may enable the method to escape a local optimum and eventually to approach 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 many data sets, for many sorts of problems. Stochastic optimization methods of this kind include:
 simulated annealing by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi (1983)[8]
 quantum annealing
 Probability Collectives by D.H. Wolpert, S.R. Bieniawski and D.G. Rajnarayan (2011)[9]
 reactive search optimization (RSO) by Roberto Battiti, G. Tecchiolli (1994),[10] recently reviewed in the reference book [11]
 crossentropy method by Rubinstein and Kroese (2004)[12]
 random search by Anatoly Zhigljavsky (1991)[13]
 Informational search [14]
 stochastic tunneling[15]
 parallel tempering a.k.a. replica exchange[16]
 stochastic hill climbing
 swarm algorithms
 evolutionary algorithms
 genetic algorithms by Holland (1975)[17]
 evolution strategies
 cascade object optimization & modification algorithm (2016)[18]
In contrast, some authors have argued that randomization can only improve a deterministic algorithm if the deterministic algorithm was poorly designed in the first place.[19] Fred W. Glover [20] argues that reliance on random elements may prevent the development of more intelligent and better deterministic components. The way in which results of stochastic optimization algorithms are usually presented (e.g., presenting only the average, or even the best, out of N runs without any mention of the spread), may also result in a positive bias towards randomness.
See also
References
 Spall, J. C. (2003). Introduction to Stochastic Search and Optimization. Wiley. ISBN 9780471330523.
 Fu, M. C. (2002). "Optimization for Simulation: Theory vs. Practice". INFORMS Journal on Computing. 14 (3): 192–227. doi:10.1287/ijoc.14.3.192.113.
 M.C. Campi and S. Garatti. The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs. SIAM J. on Optimization, 19, no.3: 1211–1230, 2008.
 Robbins, H.; Monro, S. (1951). "A Stochastic Approximation Method". Annals of Mathematical Statistics. 22 (3): 400–407. doi:10.1214/aoms/1177729586.
 J. Kiefer; J. Wolfowitz (1952). "Stochastic Estimation of the Maximum of a Regression Function". Annals of Mathematical Statistics. 23 (3): 462–466. doi:10.1214/aoms/1177729392.
 Spall, J. C. (1992). "Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation". IEEE Transactions on Automatic Control. 37 (3): 332–341. CiteSeerX 10.1.1.19.4562. doi:10.1109/9.119632.
 Holger H. Hoos and Thomas Stützle, Stochastic Local Search: Foundations and Applications, Morgan Kaufmann / Elsevier, 2004.
 S. Kirkpatrick; C. D. Gelatt; M. P. Vecchi (1983). "Optimization by Simulated Annealing". Science. 220 (4598): 671–680. Bibcode:1983Sci...220..671K. CiteSeerX 10.1.1.123.7607. doi:10.1126/science.220.4598.671. PMID 17813860.

D.H. Wolpert; S.R. Bieniawski; D.G. Rajnarayan (2011). C.R. Rao; V. Govindaraju (ed.). "Probability Collectives in Optimization". Cite journal requires
journal=
(help)  Battiti, Roberto; Gianpietro Tecchiolli (1994). "The reactive tabu search" (PDF). ORSA Journal on Computing. 6 (2): 126–140. doi:10.1287/ijoc.6.2.126.
 Battiti, Roberto; Mauro Brunato; Franco Mascia (2008). Reactive Search and Intelligent Optimization. Springer Verlag. ISBN 9780387096230.
 Rubinstein, R. Y.; Kroese, D. P. (2004). The CrossEntropy Method. SpringerVerlag. ISBN 9780387212401.
 Zhigljavsky, A. A. (1991). Theory of Global Random Search. Kluwer Academic. ISBN 9780792311225.
 Kagan E. and BenGal I. (2014). "A GroupTesting Algorithm with Online Informational Learning" (PDF). IIE Transactions, 46:2, 164184. Cite journal requires
journal=
(help)  W. Wenzel; K. Hamacher (1999). "Stochastic tunneling approach for global optimization of complex potential energy landscapes". Phys. Rev. Lett. 82 (15): 3003. arXiv:physics/9903008. Bibcode:1999PhRvL..82.3003W. doi:10.1103/PhysRevLett.82.3003.
 E. Marinari; G. Parisi (1992). "Simulated tempering: A new monte carlo scheme". Europhys. Lett. 19 (6): 451–458. arXiv:heplat/9205018. Bibcode:1992EL.....19..451M. doi:10.1209/02955075/19/6/002.
 Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. AddisonWesley. ISBN 9780201157673. Archived from the original on 20060719.
 Tavridovich, S. A. (2017). "COOMA: an objectoriented stochastic optimization algorithm". International Journal of Advanced Studies. 7 (2): 26–47. doi:10.12731/2227930x201722647.
 http://lesswrong.com/lw/vp/worse_than_random/
 Glover, F. (2007). "Tabu search—uncharted domains". Annals of Operations Research. 149: 89–98. CiteSeerX 10.1.1.417.8223. doi:10.1007/s1047900601139.
Further reading
 Michalewicz, Z. and Fogel, D. B. (2000), How to Solve It: Modern Heuristics, SpringerVerlag, New York.
 "PSA: A novel optimization algorithm based on survival rules of porcellio scaber", Y. Zhang and S. Li