In various applications, the effect of errors in gradient-based iterations is of particular importance when seeking saddle points of the Lagrangian function associated with constrained convex optimization problems. Of particular interest here are problems arising in power control applications, where network utility is maximized subject to minimum signal-to-interference-plus-noise ratio (SINR) constraints, maximum interference constraints, maximum received power constraints, or simultaneous minimum and maximum SINR constraints. Especially when the gradient iterations are executed in a distributed fashion, imperfect exchanges among the link nodes may result in erroneous gradient vectors. In order to assess and cope with such errors, two running averages (ergodic sequences) are formed from the iterates generated by the perturbed saddle point method, each with complementary strengths. Under the assumptions of problem convexity and error boundedness, bounds on the constraint violation and the suboptimality per iteration index are derived. The two types of running averages are tested on a spectrum sharing problem with minimum and maximum SINR constraints, as well as maximum interference constraints.
Bibliographical noteFunding Information:
Manuscript received August 23, 2010; revised January 17, 2011; accepted March 24, 2011. Date of publication April 19, 2011; date of current version June 15, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Mathini Sellathurai. Work in this paper was supported by NSF Grants CISE-CCF-0830480, 1016605; and ECCS-IHCS-0824007, 1002180.
- Convex programming
- error analysis
- gradient methods
- power control
- resource management