Improved Runtime Results for Simple Randomised Search Heuristics on Linear Functions with a Uniform Constraint
This work provides incremental theoretical improvements for algorithm analysis in computational optimization, relevant to researchers in evolutionary computation and randomized algorithms.
The paper tackles the problem of analyzing randomized search heuristics for linear functions with a uniform constraint, proving tight bounds on expected optimization times: Θ(n²) for RLS and improved upper bounds from O(n² log(Bw_max)) to O(n² log B) in expectation and O(n² log n) with high probability for (1+1) EA.
In the last decade remarkable progress has been made in development of suitable proof techniques for analysing randomised search heuristics. The theoretical investigation of these algorithms on classes of functions is essential to the understanding of the underlying stochastic process. Linear functions have been traditionally studied in this area resulting in tight bounds on the expected optimisation time of simple randomised search algorithms for this class of problems. Recently, the constrained version of this problem has gained attention and some theoretical results have also been obtained on this class of problems. In this paper we study the class of linear functions under uniform constraint and investigate the expected optimisation time of Randomised Local Search (RLS) and a simple evolutionary algorithm called (1+1) EA. We prove a tight bound of $Θ(n^2)$ for RLS and improve the previously best known upper bound of (1+1) EA from $O(n^2 \log (Bw_{\max}))$ to $O(n^2\log B)$ in expectation and to $O(n^2 \log n)$ with high probability, where $w_{\max}$ and $B$ are the maximum weight of the linear objective function and the bound of the uniform constraint, respectively. Also, we obtain a tight bound of $O(n^2)$ for the (1+1) EA on a special class of instances. We complement our theoretical studies by experimental investigations that consider different values of $B$ and also higher mutation rates that reflect the fact that $2$-bit flips are crucial for dealing with the uniform constraint.