Inspired by the idea of game learning, this paper studies the Cournot game model of the multi-oligopoly market and studies the realization of the Nash equilibrium by using adaptive differential evolution algorithm based on relaxation strategies. The Nash equilibrium is attained through an adaptive differential evolution based on a relaxation strategy procedure applied to an objective function with best response dynamics, the Gap function, which is derived from the sum of the payoff changes of each oligopoly under best response decisions. Furthermore, the convergence of the proposed algorithm is proved according to the random fixed point theorem and random functional theory. Finally, the calculation of a three-oligopoly game in the oil market verifies that the proposed algorithm is superior to the usual relaxation algorithm and the adaptive relaxation algorithm in terms of convergence speed and avoiding falling into the local optimal when solving multi-oligopoly Cournot game.