science

Stochastic optimization

Stochastic optimization is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Stochastic optimization rather than just read about it. In short: Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random.

Key takeaways

Stochastic optimization belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Stochastic optimization to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Stochastic optimization from memory before moving on to harder problems.

Reference excerpt

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. 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, 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) stochastic gradient descent finite-difference SA by Kiefer and Wolfowitz (1952) simultaneous perturbation SA by Spall (1992) scenario optimization

Randomized search methods

On the other hand, even when the data set consists of precise measurements, some methods introduce randomness into the search-process to accelerate progress. 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. 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) quantum annealing Probability Collectives by D.H. Wolpert, S.R. Bieniawski and D.G. Rajnarayan (2011) reactive search optimization (RSO) by Roberto Battiti, G. Tecchiolli (1994), recently reviewed in the reference book cross-entropy method by Rubinstein and Kroese (2004) random search by Anatoly Zhigljavsky (1991) Informational search stochastic tunneling parallel tempering a.k.a. replica exchange stochastic hill climbing swarm algorithms evolutionary algorithms genetic algorithms by Holland (1975) evolution strategies cascade object optimization & modification algorithm (2016) 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. Fred W. Glover 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

References

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

External links COSP

Worked examples

Example 1 — a first encounter with Stochastic optimization

Start with the simplest possible case. Write down what Stochastic optimization claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Stochastic optimization before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Stochastic optimization ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Stochastic optimization

In research: Stochastic optimization appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Stochastic optimization in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Stochastic optimization is common in secondary-school and first-year university syllabi. It links to neighbouring topics Monte Carlo methods, Stochastic optimization, so understanding it makes those chapters shorter.
In everyday life: Look for Stochastic optimization outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Stochastic optimization in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Stochastic optimization means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Stochastic optimization out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stochastic optimization in simple terms?

Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random.

Why does Stochastic optimization matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Stochastic optimization?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Stochastic optimization.