PENALTY BARRIER MULTIPLIER METHODS FOR CONVEX-PROGRAMMING PROBLEMS/

We study a class of methods for solving convex programs, which are bas ed on non-quadratic augmented Lagrangians for which the penalty parame ters are functions of the multipliers. This gives rise to Lagrangians which are nonlinear in the multipliers. Each augmented Lagrangian is s pecified by a choice of a penalty function phi and a penalty-updating function pi. The requirements on pi are mild and allow for the inclusi on of most of the previously suggested augmented Lagrangians. More imp ortantly, a new type of penalty/barrier function (having a logarithmic branch glued to a quadratic branch) is introduced and used to constru ct an efficient algorithm. Convergence of the algorithms is proved for the case of pi being a sublinear function of the dual multipliers. Th e algorithms are tested on large-scale quadratically constrained probl ems arising in structural optimization.