Primal-Dual Subgradient Method for Huge-Scale Linear Conic Problems

In this paper we develop a primal-dual subgradient method for solving huge-scale Linear Conic Optimization Problems. Our main assumption is that the primal cone is formed as a direct product of many small-dimensional convex cones, and that the matrix A of corresponding linear operator is uniformly sparse. In this case, our method can approximate the primal-dual optimal solution with accuracy ε in O ( 1 ε2 ) iterations. At the same time, complexity of each iteration of this scheme does not exceed O(rq log2 n) operations, where r and q are the maximal numbers of nonzero elements in the rows and columns of matrix A, and n is the number variables.