It sounds like you're referring to something like simulated annealing. Using that as an example, the fundamental requirement is to introduce arbitrary, uncorrelated steps -- there's no requirement that the steps be random, and the only potential advantage of using a random source is that it provides independence (lack of correlation) inherently; but in exchange, it makes testing and reproduction much harder. Basically every use of simulated annealing or similar I've run into uses pseudorandom numbers for this reason.