Studies on continuous and discrete structures in optimization and game theory

我们将三相划分问题的对偶定理推广到两种情况。一种是最优解是退化的情况，另一种是在高维空间中。我们给出了两类离散不动点定理。一个是单调映射，另一个是收缩映射。我们证明了在二人博弈中单调性是必要的。进一步，在browwer不动点定理的基础上，我们删除了离散不动点定理中的凸性假设。

We have extended a duality theorem for the three-phase partition problem to two cases. One is the case that the optimal solution is degenerate, and the other is in higher dimensional space. We have given two types of discrete fixed point theorems. One is for monotone mappings and the other is for contraction mappings. We showed that the monotonicity is necessary in the two-person game. Further, we delete convexity assumption from discrete fixed point theorem based on Brouwer's fixed point theorem.