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This work examines the expected computational cost to determine anapproximate global minimum of a class of cost functions characterized by thevariance of coefficients. The cost function takes $N$-dimensional binary statesas arguments and has many local minima. Iterations in the order of $2^N$ arerequired to determine an approximate global minimum using random search. Thiswork analytically and numerically demonstrates that the selection and crossoveralgorithm with random initialization can reduce the required computational cost(i.e., number of iterations) for identifying an approximate global minimum tothe order of $\lambda^N$ with $\lambda$ less than 2. The two best solutions,referred to as parents, are selected from a pool of randomly sampled states.Offspring generated by crossovers of the parents' states are distributed with amean cost lower than that of the original distribution that generated theparents. It is revealed that in contrast to the mean, the variance of the costof the offspring is asymptotically the same as that of the originaldistribution. Consequently, sampling from the offspring's distribution leads toa higher chance of determining an approximate global minimum than sampling fromthe original distribution, thereby accelerating the global search. This featureis distinct from the distribution obtained by a mixture of a large populationof favorable states, which leads to a lower variance of offspring. Thesefindings demonstrate the advantage of the crossover between two favorablestates over a mixture of many favorable states for an efficient determinationof an approximate global minimum.

Computational cost for determining an approximate global minimum using the selection and crossover algorithm