Multiuser Optimization: Distributed Algorithms and Error Analysis

Traditionally, a multiuser problem is a constrained optimization problem characterized by a set of users, an objective given by a sum of user-specific utility functions, and a collection of linear constraints that couple the user decisions. The users do not share the information about their utilities, but do communicate values of their decision variables. The multiuser problem is to maximize the sum of the users-specific utility functions subject to the coupling constraints, while abiding by the informational requirements of each user. In this paper, we focus on generalizations of convex multiuser optimization problems where the objective and constraints are not separable by user and instead consider instances where user decisions are coupled, both in the objective and through nonlinear coupling constraints. To solve this problem, we consider the application of gradient-based distributed algorithms on an approximation of the multiuser problem. Such an approximation is obtained through a Tikhonov regularization and is equipped with estimates of the difference between the optimal function values of the original problem and its regularized counterpart. In the algorithmic development, we consider constant steplength primal, primal-dual and dual schemes in which the iterate computations are distributed naturally across the users, i.e., each user updates its own decision only. The primal scheme, of relevance when user decisions are uncoupled, is presented along with per-iteration error bounds for regimes where communication failures across users may occur. When user decisions are coupled, we consider primal-dual and dual schemes. Convergence theory in the primal-dual space is provided in limited coordination settings, and allows for differing steplengths across users as well as across the primal and dual space. An alternative to primal-dual schemes can be found in dual schemes which are analyzed in regimes where primal solutions are obtained through a fixed number of gradient steps. Our results are supported by a case-study in which the proposed algorithms are applied to a multi-user problem arising in a congested traffic network.