NUMERICAL STABILITY AND EFFICIENCY OF PENALTY ALGORITHMS

Penalty algorithms have been somewhat forgotten due to numerical insta bilities once believed to be inherent to those methods. One usually ha s to solve a sequence of such problems, and when the penalty factor de creases too fast, the subproblems may become intractable. Moreover, as the penalty factor decreases, the unconstrained subproblem becomes il l conditioned, and thus difficult to solve. Also, in several intermedi ate computations, numerical instability may show up. The author propos es remedies to such problems and presents a wide class of numerically stable penalty algorithms. The work is done in the more general contex t of variational inequality problems, which encompasses optimization p roblems. The author's results yield a family of globally convergent, t wo-step superlinearly convergent, numerically stable algorithms for va riational inequality problems. Finally, issues in the numerically stab le implementation of intermediate computations within those algorithms are discussed.