# Which optimization algorithm to use for problems with many local optima and expensive goal function?

I've got an optimization problem with these properties:

• The goal function is not cheap to compute. It can be evaluated up to around 10^4 times in the optimization.
• There's a lot of local optima.
• High values are clustered around the maximum, so the problem is kind of convex.
• The solutions are constrained to a known hyperbox.
• The gradient is unknown, but intuitively, the function is smooth.
• It's a function of up to 50 variables. The graphs below are examples of the 2D special case.

Nelder-Mead gets stuck in the local optima a lot. Above three variables, there's no chance of using brute force. I've looked into genetic algorithms, but they seem to require a lot of evaluations of the goal function.

What kind of algorithm should I be looking for?

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Anna, this is potentially interesting here on the stats site--because as you know, many statistical procedures look for optima of functions like this--but can you draw an explicit connection with a statistical problem to convince the community not to vote to close your question as off-topic? –  whuber Jun 11 '12 at 18:49
@whuber: I actually thought questions on numeric optimization were on-topic. The goal function is a measure of different correlation matrices, but there is no clear statistical connection. Is there a better site to ask this question on? –  Anna Jun 11 '12 at 18:57
I am not aware of a better site, Anna. I believe the difficulty is that the field of optimization is far broader than stats or machine learning; it has its own literature, terminology, methods, and so on. Statisticians become acquainted with some optimization concepts and methods perforce, but it is rarely their specialty or primary interest. –  whuber Jun 11 '12 at 19:00
@Procrastinator SA typically requires between $10^4$ and $10^5$ (or more) evaluations during its random searching before it even begins to "cool" near a reasonable optimum. It's great for functions whose differences can be very rapidly evaluated in terms of a small to medium-size proposed change in their arguments. (This can include functions that are expensive to evaluate ab initio, if one is both lucky and clever.) Otherwise, for expensive functions, SA may be one of the worst possible approaches. –  whuber Jun 12 '12 at 11:46
@Proc stackovergio has given a good reply (+1): scan the "Efficient Global Optimization" paper. It reports impressive results with as few as 20 evaluations in some (quasi-realistic) cases. It's hard to recommend anything specific for this question because we aren't told much about the nature of the function or its domain of definition. –  whuber Jun 12 '12 at 12:01

In case of expensive functions without derivative, a useful abstract framework for optimization is

Compute the function in few points (it may be a regular grid or even not)
Repeat
Interpolate data/fit into a stochastic model
Validate the model through statistical tests
Find the model’s maximum.
If it is better that the best one you previously have got, update the maximum
Put this point in the dataset


That should also fit well with the pseudo convexity you refered.

References here:

Efficient Global Optimization of Expensive Black-Box Functions

A Rigorous Framework for Optimization of Expensive Functions by Surrogates

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