  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, a tractable approximated DRCP (ADRCP) is introduced by
discretizing the semi-infinite constraints into a finite number of inequality
constraints and restricting the right-hand side of the constraints with a
positive parameter. This problem is iteratively solved by a distributed
projected gradient algorithm proposed in this paper, which is based on
epigraphic reformulation and subgradient projected algorithms. Secondly, a
cutting-surface consensus approach is proposed for locating an approximately
optimal consensus solution of the DRCP with guaranteed feasibility. This
approach is based on iteratively approximating the DRCP by successively
reducing the restriction parameter of the right-hand constraints and populating
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.
