An adaptive barrier method for convex programming

This paper presents a new barrier method for convex programming. The method involves an optimization transfer principle. Instead of minimizing the objective function f(x) directly, one minimizes the amended function f(x) − μ P i x i ln xi to produce the next iterate x n+1 from the current iterate x. If the feasible region is contained in the unit simplex, then this strategy forces a decrease in f(x). The barrier parameter μ is kept constant during the process and not sent gradually to 0 as in the classical barrier method. Under mild assumptions on f(x) and the linear constraints, the method converges to the global minimum of f(x). If this minimum occurs in the interior of the feasible region, then the rate of convergence is linear.