Cooperative Optimization for Inseparable Objective Functions with Coupled Inequality Constraints

The task of cooperative optimization in a multi-agent system is that multiple agents work together to achieve a global optimization goal. Coupled constraints on behaviors of agents undoubtedly make this task more difficult. In this paper, we study the cooperative optimization problem with coupled inequality constraints and develop a design framework based on potential game theory to solve this problem. Different from the existing results, we do not require the objective function to be the summation form of local utilities, nor do we require the coupled constraints to be linear or nonlinear. We use logarithmic barrier method to transform the constrained optimization problem to an unconstrained optimization problem. A potential game is set up in Euclidean space so that the unconstrained optimization problem can be solved in a distributed way. We also prove that the solution of the decoupled optimization problem is equivalent to the solution of the original problem. An improved gradient projection algorithm is proposed and its convergence is illustrated through a numerical example.