Final answer:
The largest possible number of distinct local optima, assuming each of the three problem solvers correctly identifies the global optimum and each has five local optima, would be 13. This accounts for the overlap of the global optimum across all solvers' sets, which is only counted once.
Step-by-step explanation:
Given that we have three problem solvers, each with five local optima and assuming that each problem solver correctly identifies the global optimum, the question is: what is the largest possible number of distinct local optima? Since each solver's local optima may overlap with others—or could potentially be identical—we must consider the different scenarios.
Assuming none of the local optima overlap and all are unique, each solver contributes five distinct local optima to the overall set.
However, if the global optimum is the same for all three solvers, this optimum would not be counted triple times. Therefore, the maximum number of distinct (non-overlapping) local optima would be the sum of the local optima identified by all solvers minus two (since the global optimum is counted just once).
If each problem solver's set of local optima is unique, the answer is simple. Each local optimum contributes uniquely to the overall set, and we can just add them up:
5 local optima per solver x 3 solvers = 15 distinct local optima
However, we must subtract the global optimum that is counted once, but would have been included in each solver's set of five, thus:
15 distinct local optima - 2 = 13 distinct local optima
This would be the maximum number of distinct local optima possible in this case.