Co-evolutionary algorithms (CoEAs), which pair candidate designs with test cases, are frequently used in adversarial optimisation, particularly for binary test-based problems where designs and tests yield binary outcomes. The effectiveness of designs is determined by their performance against tests, and the value of tests is based on their ability to identify failing designs, often leading to more sophisticated tests and improved designs. However, CoEAs can exhibit complex, sometimes pathological behaviours like disengagement. Through runtime analysis, we aim to rigorously analyse whether CoEAs can efficiently solve test-based adversarial optimisation problems in an expected polynomial runtime. This paper carries out the first rigorous runtime analysis of $(1,\lambda)$ CoEA for binary test-based adversarial optimisation problems. In particular, we introduce a binary test-based benchmark problem called \Diagonal problem and initiate the first runtime analysis of competitive CoEA on this problem. The mathematical analysis shows that the $(1,\lambda)$-CoEA can efficiently find an $\varepsilon$ approximation to the optimal solution of the \Diagonal problem, i.e. in expected polynomial runtime assuming sufficiently low mutation rates and large offspring population size. On the other hand, the standard $(1,\lambda)$-EA fails to find an $\varepsilon$ approximation to the optimal solution of the \Diagonal problem in polynomial runtime. This suggests the promising potential of coevolution for solving binary adversarial optimisation problems.

翻译：协同进化算法（CoEAs）将候选设计与测试用例配对，常用于对抗优化，特别是在二元测试问题中，其中设计与测试产生二元结果。设计的有效性取决于其在测试中的表现，而测试的价值则基于其识别失败设计的能力，这通常会导致更复杂的测试和改进的设计。然而，协同进化算法可能表现出复杂、有时甚至是病态的行为，如脱离。通过运行时分析，我们旨在严格分析协同进化算法是否能在预期的多项式运行时内高效解决基于测试的对抗优化问题。本文首次对基于二元测试的对抗优化问题进行了$(1,\lambda)$协同进化算法的严格运行时分析。具体而言，我们引入了一个名为\Diagonal问题的二元测试基准问题，并首次对该问题上的竞争性协同进化算法进行了运行时分析。数学分析表明，$(1,\lambda)$-协同进化算法能够高效地找到\Diagonal问题最优解的$\varepsilon$近似解，即在假设足够低的突变率和较大的后代种群规模下，在预期的多项式运行时内实现。另一方面，标准的$(1,\lambda)$-进化算法无法在多项式运行时内找到\Diagonal问题最优解的$\varepsilon$近似解。这表明协同进化在解决二元对抗优化问题方面具有广阔潜力。