The Runtime of Random Local Search on the Generalized Needle Problem
This work addresses a theoretical gap in evolutionary computation by rigorously analyzing runtime complexities for a specific optimization problem, though it is incremental as it builds on prior bounds.
The authors derived an exact description of the expected runtime for randomized local search on generalized Needle functions, significantly improving previous upper bounds and providing asymptotic estimates to determine the influence of the needle radius parameter.
In their recent work, C. Doerr and Krejca (Transactions on Evolutionary Computation, 2023) proved upper bounds on the expected runtime of the randomized local search heuristic on generalized Needle functions. Based on these upper bounds, they deduce in a not fully rigorous manner a drastic influence of the needle radius $k$ on the runtime. In this short article, we add the missing lower bound necessary to determine the influence of parameter $k$ on the runtime. To this aim, we derive an exact description of the expected runtime, which also significantly improves the upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates of the expected runtime.