Yeah, looking at it again it's not as accessible as I remember. It's quite mathy, but well-written. An epsilon-ball is just a "ball" (circle or sphere analogue

Yeah, looking at it again it's not as accessible as I remember. It's quite mathy, but well-written. An epsilon-ball is just a "ball" (circle or sphere analogue of dimension n, where n is the number of inputs) of radius epsilon (some arbitrarily small number). So you're correct that it relates to the resolution at which we can be said to "know" the solution. With simulated annealing, the cooling schedule dictates how and when the algorithm switches from exploration to exploitation. With a logarithmic cooling schedule, the cooling rate is 1 / log (1 + t), so the temperature approaches zero extremely slowly, and the algorithm basically never switches out of exploration mode. You're correct that this is basically a brute force search of the domain, and this is the implication of the theorem I mention above--ANY global optimizer reduces to brute force search in the worst case.