The distributed task allocation problem, as one of the most interesting distributed optimization challenges, has received considerable research attention recently. Previous works mainly focused on the task allocation problem in a population of individuals, where there are no constraints for affording task amounts. The latter condition, however, cannot always be hold. In this paper, we study the task allocation problem with constraints of task allocation in a game-theoretical framework. We assume that each individual can afford different amounts of task and the cost function is convex. To investigate the problem in the framework of population games, we construct a potential game and calculate the fitness function for each individual. We prove that when the Nash equilibrium point in the potential game is in the feasible solutions for the limited task allocation problem, the Nash equilibrium point is the unique globally optimal solution. Otherwise, we also derive analytically the unique globally optimal solution. In addition, in order to confirm our theoretical results, we consider the exponential and quadratic forms of cost function for each agent. Two algorithms with the mentioned representative cost functions are proposed to numerically seek the optimal solution to the limited task problems. We further perform Monte Carlo simulations which provide agreeing results with our analytical calculations.

翻译：分布式任务分配问题作为最具趣味的分布式优化挑战之一，近年来获得了广泛的研究关注。以往的研究主要关注个体群体中的任务分配问题，其中不存在任务量承载能力的约束。然而，后一条件并非总能成立。本文在博弈论框架下研究具有任务分配约束的问题。我们假设每个个体可承担不同任务量，且成本函数为凸函数。为在种群博弈框架下探究该问题，我们构建了一个势博弈模型，并计算每个个体的适应度函数。我们证明：当势博弈中的纳什均衡点属于有限任务分配问题的可行解集时，该纳什均衡点即为唯一的全局最优解。否则，我们同样通过解析方法推导出唯一的全局最优解。此外，为验证理论结果，我们考虑了每个智能体的指数形式和二次形式的成本函数。基于上述两种代表性成本函数，提出两种算法以数值方式求解有限任务问题的最优解。进一步进行蒙特卡洛模拟，所得结果与解析计算结果一致。