Minimization Algorithms and Random Walk on the $d$-Cube

Abstract

Consider the number of steps needed by algorithms to locate the minimum of functions defined on the $d$-cube, where the functions are known to have no local minima except the global minimum. Regard this as a game: one player chooses a function, trying to make the number of steps needed large, while the other player chooses an algorithm, trying to make this number small. It is proved that the value of this game is approximately of order $2^{d/2}$ steps as $d \rightarrow \infty$. The key idea is that the hitting times of the random walk provide a random function for which no algorithm can locate the minimum within $2^{d(1/2 - \varepsilon)}$ steps.