In the previous work, the stability condition of the particle swarm optimization algorithm has been derived by linear matrix inequality techniques. This paper provides an alternative representation of the stability condition, which can be checked more shortly and accurately than that in the previous work. The stability condition is described by nonlinear scalar inequalities. Also, we present a condition related to a decay rate of the particle swarm optimization algorithm in terms of nonlinear scalar inequalities. Numerical experiments are given to show that the largest lower bound of the decay rate can be used as a measure of convergence speed of the particle swarm optimization algorithm.

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Each finite zero-sum two-person game is defined by the numbers, m and n, of possible strategies of two players, and the payoff a_ij (from the second player P₂ to the first player P₁) corresponding to the strategies i and j chosen respectively by P₁ and P₂. In this paper, we consider the case where each element a_ij is a random variable and discuss stability of the value of the game denoted by {v_mn} as m → ∞ and n → ∞. The main stability result is derived from the law of large numbers for random linear programming problems. An upper bound of game values is also obtained as the maximum loss of player P₂ by choosing a uniform, purely random strategy.

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