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The minimisation of cost functions is crucial in various optimisation fields.However, identifying their global minimum remains challenging owing to the hugecomputational cost incurred. This work analytically expresses the computationalcost to identify an approximate global minimum for a class of cost functionsdefined under a high-dimensional discrete state space. Then, we derive anoptimal global search scheme that minimises the computational cost.Mathematical analyses demonstrate that a combination of the gradient descentalgorithm and the selection and crossover algorithm--with a biased crossoverweight--maximises the search efficiency. Remarkably, its computational cost isof the square root order in contrast to that of the conventional gradientdescent algorithms, indicating a quadratic speedup of global search. Wecorroborate this proposition using numerical analyses of the travellingsalesman problem. The simple computational architecture and minimalcomputational cost of the proposed scheme are highly desirable for biologicalorganisms and neuromorphic hardware.

Quadratic speedup of global search using a biased crossover of two good solutions