Stochastic optimization explained

Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random. Stochastic optimization also include methods with random iterates. Some hybrid 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 real-time estimation and control, simulation-based 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 finite-difference SA by Kiefer and Wolfowitz (1952)^[5]
• simultaneous perturbation SA by Spall (1992)^[6]
• scenario optimization

Randomized search methods

See also: Metaheuristic. On the other hand, even when the data set consists of precise measurements, some methods introduce randomness into the search-process to accelerate progress.^[7] Such randomness can also make the method less sensitive to modeling errors. Another advantage is that randomness into the search-process can be used for obtaining interval estimates of the minimum of a function via extreme value statistics.^{[8] [9]} 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)^[10]
• quantum annealing
• Probability Collectives by D.H. Wolpert, S.R. Bieniawski and D.G. Rajnarayan (2011)^[11]
• reactive search optimization (RSO) by Roberto Battiti, G. Tecchiolli (1994),^[12] recently reviewed in the reference book ^[13]
• cross-entropy method by Rubinstein and Kroese (2004)^[14]
• random search by Anatoly Zhigljavsky (1991)^[15]
• Informational search ^[16]
• stochastic tunneling^[17]
• parallel tempering a.k.a. replica exchange^[18]
• stochastic hill climbing
• swarm algorithms
• evolutionary algorithms
  • genetic algorithms by Holland (1975)^[19]
  • evolution strategies
• cascade object optimization & modification algorithm (2016)^[20]

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.^[21] Fred W. Glover^[22] 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

• Global optimization
• Machine learning
• Scenario optimization
• Gaussian process
• State Space Model
• Model predictive control
• Nonlinear programming
• Entropic value at risk

Further reading

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

External links

• COSP

Notes and References

1. Book: Spall, J. C. . Introduction to Stochastic Search and Optimization . 2003 . Wiley . 978-0-471-33052-3 .
2. Fu, M. C. . Optimization for Simulation: Theory vs. Practice . INFORMS Journal on Computing . 2002 . 14 . 192–227 . 10.1287/ijoc.14.3.192.113 . 3.
3. 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.http://epubs.siam.org/siopt/resource/1/sjope8/v19/i3/p1211_s1
4. Robbins, H. . Monro, S. . A Stochastic Approximation Method . Annals of Mathematical Statistics . 1951 . 22 . 400–407 . 10.1214/aoms/1177729586 . 3. free .
5. J. Kiefer . Jack Kiefer (mathematician) . J. Wolfowitz . Jacob Wolfowitz . Stochastic Estimation of the Maximum of a Regression Function . Annals of Mathematical Statistics . 1952 . 23 . 462–466 . 10.1214/aoms/1177729392 . 3. free .
6. Spall, J. C. . Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation . IEEE Transactions on Automatic Control . 1992 . 37 . 332–341 . 10.1109/9.119632 . 3. 10.1.1.19.4562 .
7. Holger H. Hoos and Thomas Stützle, Stochastic Local Search: Foundations and Applications, Morgan Kaufmann / Elsevier, 2004.
8. M. de Carvalho . Confidence intervals for the minimum of a function using extreme value statistics . International Journal of Mathematical Modelling and Numerical Optimisation . 2 . 2011 . 3 . 288–296 . 10.1504/IJMMNO.2011.040793 .
9. M. de Carvalho . A generalization of the Solis-Wets method . Journal of Statistical Planning and Inference . 142 . 2012 . 3 . 633‒644 . 10.1016/j.jspi.2011.08.016 .
10. S. Kirkpatrick . C. D. Gelatt . M. P. Vecchi . Optimization by Simulated Annealing . Science . 1983 . 220 . 671–680 . 10.1126/science.220.4598.671 . 17813860 . 4598. 1983Sci...220..671K . 10.1.1.123.7607 . 205939 .
11. Web site: D.H. Wolpert . S.R. Bieniawski . D.G. Rajnarayan . Probability Collectives in Optimization . 2011. Santa Fe Institute .
12. Battiti. Roberto. Gianpietro Tecchiolli. 1994. The reactive tabu search. ORSA Journal on Computing. 6. 2. 126–140. 10.1287/ijoc.6.2.126.
13. Book: Battiti , Roberto . Reactive Search and Intelligent Optimization. Mauro Brunato . Franco Mascia . 2008. Springer Verlag. 978-0-387-09623-0.
14. Book: Rubinstein, R. Y. . Reuven Rubinstein . Kroese, D. P. . Dirk Kroese . The Cross-Entropy Method . 2004 . Springer-Verlag . 978-0-387-21240-1.
15. Book: Zhigljavsky, A. A. . Theory of Global Random Search . 1991 . Kluwer Academic . 978-0-7923-1122-5.
16. A Group-Testing Algorithm with Online Informational Learning. Kagan E. . Ben-Gal I. . IIE Transactions . 46 . 2 . 164–184. 2014. 10.1080/0740817X.2013.803639. 18588494 .
17. W. Wenzel . K. Hamacher . Stochastic tunneling approach for global optimization of complex potential energy landscapes . Phys. Rev. Lett. . 82 . 1999 . 3003 . 10.1103/PhysRevLett.82.3003. 1999PhRvL..82.3003W . 15. physics/9903008 . 5113626 .
18. E. Marinari . G. Parisi . Simulated tempering: A new monte carlo scheme . Europhys. Lett. . 19 . 1992 . 451–458 . 10.1209/0295-5075/19/6/002 . 6. hep-lat/9205018 . 1992EL.....19..451M . 12321327 .
19. Book: Goldberg, D. E. . Genetic Algorithms in Search, Optimization, and Machine Learning . 1989 . Addison-Wesley . 978-0-201-15767-3 . https://web.archive.org/web/20060719133933/http://www-illigal.ge.uiuc.edu/ . 2006-07-19.
20. Tavridovich, S. A. . COOMA: an object-oriented stochastic optimization algorithm . International Journal of Advanced Studies . 7 . 2017 . 26–47 . 2 . 10.12731/2227-930x-2017-2-26-47. free .
21. Web site: Worse Than Random - LessWrong . Yudkowsky . Eliezer .
22. Glover, F. . 2007 . Tabu search—uncharted domains . Annals of Operations Research . 149 . 89–98 . 10.1007/s10479-006-0113-9. 10.1.1.417.8223 . 6854578 .