A Random Coordinate Descent Algorithm for Singly Linear Constrained Smooth Optimization∗

In this paper we develop a novel randomized block-coordinate descent method for minimizing multi-agent convex optimization problems with singly linear coupled constraints over networks and prove that it obtains in expectation an ε accurate solution in at most O( 1 λ2(Q)ε ) iterations, where λ2(Q) is the second smallest eigenvalue of a matrix Q that is defined in terms of the probabilities and the number of blocks. However, the computational complexity per iteration of our method is much simpler than of a method based on full gradient information and each iteration can be computed in a completely distributed way. We focus on how to choose the probabilities to make this randomized algorithm to converge as fast as possible and we arrive at solving a sparse SDP. Numerical tests confirm that on huge optimization problems our method is much more numerically efficient than methods based on full gradient.