Black-box combinatorial optimization problem over permutations

Question

I am solving general black-box optimization problems like: x*: f(x) -> min, where x are permutations of length N (N = 50 for example, so brute force search is not possible). Objective function f(x) is represented by stand-alone computer code and x represents configuration of complex system with the response simulated by f(x).

I learned, that in this case I can use many heuristic methods. But, most of these methods use always some kind of local search, which require suitable distance metric at search space (space of permutations x in my case). Under suitable distance metric I mean the metric which fulfill the "locality" property, e.g. small change of permutation x produce small change of objective function f(x). In my case is not known any suitable distance metric with this property, so any kind of local search is nearly the random search.

I have a few questions:

1. Are there available any heuristic black-box combinatorial optimization methods, which does not use local search and/or any distance metric at search space? I need to overcome the low "locality" of the problem or simply the fact, that any suitable distance metric at search space is unknown.

2. Is the "locality" property really so restricted at combinatorial optimization in general? May be I miss something..., but the most of real-world black-box combinatorial problem has low or very low "locality" due to the fact, that the common permutation distance metrics (Hamming, Kendal, etc.) are not suitable metrics in general.

3. Is there any general method how to find suitable distance metric at search space to satisfy at least approximately the "locality"?

Additional remarks:

• In real, the black-box function f(x) is realized by stand-alone deterministic simulation code, where x plays a role of discrete configuration of the simulated physical system. So, function f(x) has definitely well defined properties, but this properties are so difficult, that is not possible to simple exploit it.

• Because of above mentioned complicated internal properties of function f(x) is not possible to find proper distance metric d(x,x') in search space which fulfill "locality" (similar x and x' in a sense of any distance metric produce similar responses f(x) and f(x'))

• So, finally, I am looking for any optimization heuristics, which are able to find any suitable sub-optimal solutions only by informations available by properties of f(x) at fitness space. Like EDA's (Estimation of Distribution Algorithms) for example.

The main reason of this question is, what types of optimization heuristics are suitable to solve this kind of problems.

Answer 1

In general if you know nothing about f and it can be totally arbitrary, then there is nothing you can do: you cannot do better than brute-force. If f has no structure, no properties, no regularities, and is just totally random, then it is easy to see that any algorithm has to compute f on all $N!$ possible permutations to find the minimum.

In practice it is common to deal with functions f that do have some degree of structure. For instance, if x,x' are similar (for some notion of similarity), then f(x),f(x') might have a good chance of being similar. In those cases you can use techniques like local search. But if you don't have that property, then you can't use local search, and you may be back to a situation where you cannot do any better than brute force.

There are other approaches in other situations: e.g., if we know the symbolic expression for f, we might be able to use a SAT solver, especially if that symbolic expression is not too large. But again this requires f to have some structure and not be totally arbitrary.

There are no silver bullets. You need to know something about f to have a hope of doing anything.