  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.
