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 $$\varTheta (n^2)$$ 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.

中文翻译：

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改进了具有统一约束的线性函数的简单随机搜索启发式的运行时结果

在过去十年中，在开发用于分析随机搜索启发式的合适证明技术方面取得了显着进展。对这些函数类算法的理论研究对于理解潜在的随机过程至关重要。传统上一直在该领域研究线性函数，导致此类问题的简单随机搜索算法的预期优化时间有严格的界限。最近，这个问题的约束版本引起了人们的关注，并且在这类问题上也取得了一些理论成果。在本文中，我们研究了均匀约束下的线性函数类，并研究了随机局部搜索 (RLS) 和称为 (1+1) EA 的简单进化算法的预期优化时间。我们证明了 RLS 的 $$\varTheta (n^2)$$ 的紧边界，并从 $$O(n^2 \log (Bw_{\max } ))$$ 到 $$O(n^2\log B)$$ 的期望值和 $$O(n^2 \log n)$$ 的概率很高，其中 $$w_{\max }$$ 和B 分别是线性目标函数的最大权重和统一约束的边界。此外，我们在特殊类别的实例上为 (1+1) EA 获得了严格的 $$O(n^2)$$ 边界。我们通过考虑不同 B 值和更高突变率的实验研究补充了我们的理论研究，这反映了 2 位翻转对于处理均匀约束至关重要的事实。