A systemAn optimization process is performing optimization if it is movingany kind of process that systematically comes up with solutions that are better than the solution used before. More technically, this kind of process moves the world into a specific and unexpected set of states.states by searching through a large search space, hitting small and low probability targets. When this process is gradually guided by some agent into some specific state, through searching specific targets, we can say it prefers that state.
The best way to exemplify an optimization process is through a simple example: Eliezer Yudkowsky suggests natural selection is such a process. Through an implicit preference – better replicators – natural selection searches all the genetic landscape space and hit small targets: efficient mutations.
Consider the human being. We are a highly complex object with a low probability to have been created by chance - natural selection, however, over millions of years, built up the infrastructure needed to build such a functioning body. This body, as well as other organisms, had the chance (was selected) to develop because it is in itself a rather efficient replicator suitable for the environment where it came up.
Or consider the famous chessplaying computer, Deep Blue. Outside of the narrow domain of selecting moves for chess games, it can't do anything impressive: but as a chessplayer, it was massively more effective than virtually all humans. It has a high optimization power in the chess domain but almost none in any other field. Humans or evolution, on the other hand, are more domain-general optimization processes than Deep Blue, but that doesn't mean they're more effective at chess specifically. (Although note in what contexts this optimization process abstraction is useful and where it fails to be useful: it's not obvious what it would mean for "evolution" to play chess, and yet it is useful to talk about the optimization power of natural selection, or of Deep Blue.)
One way to think mathematically about optimization, like evidence, is in information-theoretic bits. The optimization power is the amount of surprise we would have in the result if there were no optimization process present. Therefore we take the base-two logarithm of the reciprocal of the probability of the result. A one-in-a-million solution (a solution so good relative to your preference ordering that it would take a million random tries to find something that good or better) can be said to have log_2(1,000,000) = 19.9 bits of optimization. Compared to a random configuration of matter, any artifact you see is going to be much more optimized than this. The math describes only laws and general principles for reasoning about optimization; as with probability theory, you oftentimes can't apply the math directly.
A system is performing optimization if it is moving the world into a specific and unexpected set of states.