Penalty/barrier multiplier methods for convex programming problems

We study a class of methods for solving convex programs, which are based on non-quadratic augmented Lagrangians for which the penalty parameters are functions of the multipliers. This gives rise to Lagrangians which are nonlinear in the multipliers. Each augmented Lagrangian is specified by a choice of a penalty function phi and a penalty-updating function pi. The requirements on pi are mild and allow for the inclusion of most of the previously suggested augmented Lagrangians. More importantly, a new type of penalty/barrier function (having a logarithmic branch glued to a quadratic branch) is introduced and used to construct an efficient algorithm. Convergence of the algorithms is proved for the case of pi being a sublinear function of the dual multipliers. The algorithms are tested on large-scale quadratically constrained problems arising in structural optimization.