Stochastic Composite Optimization & Game: Modeling Cooperation-Competition in Uncertain Environments

From the perspective of the relationship between the number of groups and players, the problem 1 can be considered as including two categories: stochastic composite optimization problem and stochastic composite game problem. Specifically, when the number of groups is 1, the competition between groups will disappear, and problem 1 degenerates into a stochastic composite optimization problem where multiple agents cooperate to minimize an objective function. Similarly, when the number of agents within each group is 1, the cooperation within the group will be abandoned, and problem 1 is transformed into a situation where multiple self-interested agents compete, namely a composite game problem. Therefore, problem 1 not only models the cooperation-competition relationship between agents in an uncertain environment, but also includes discussions on distributed optimization problems and distributed game problems in general situations.