A cutting-surface consensus approach for distributed robust optimization of multi-agent systems

A novel and fully distributed optimization method is proposed for the distributed robust convex program (DRCP) over a time-varying unbalanced directed network under the uniformly jointly strongly connected (UJSC) assumption. Firstly, an approximated DRCP (ADRCP) is introduced by discretizing the semi-infinite constraints into a finite number of inequality constraints to ensure tractability and restricting the right-hand side of the constraints with a positive parameter to ensure a feasible solution for (DRCP) can be obtained. This problem is iteratively solved by a distributed projected gradient algorithm proposed in this paper, which is based on epigraphic reformulation and gradient projected operations. Secondly, a cutting-surface consensus approach is proposed for locating an approximately optimal consensus solution of the DRCP with guaranteed local feasibility for each agent. This approach is based on iteratively approximating the DRCP by successively reducing the restriction parameter of the right-hand constraints and adding the cutting-surfaces into the existing finite set of constraints. Thirdly, to ensure finite-time termination of the distributed optimization, a distributed termination algorithm is developed based on consensus and zeroth-order stopping conditions under UJSC graphs. Fourthly, it is proved that the cutting-surface consensus approach terminates finitely and yields a feasible and approximate optimal solution for each agent. Finally, the effectiveness of the approach is illustrated through a numerical example.