On controlling the parameter in the logarithmic barrier term for convex programming problems

We present a log-barrier based algorithm for linearly constrained convex differentiable programming problems in nonnegative variables, but where the objective function may not be differentiable at points having a zero coordinate. We use an approximate centering condition as a basis for decreasing the positive parameter of the log-barrier term and show that the total number of iterations to achieve an -tolerance optimal solution isO(|log()|)×(number of inner-loop iterations). When applied to then-variable dual geometric programming problem, this bound becomesO(n ² U/), whereU is an upper bound on the maximum magnitude of the iterates generated during the computation.The authors gratefully acknowledge very constructive and insightful comments and suggestions from the two anonymous referees and the correspondence from A. V. Fiacco (Ref. 1).