Penalty and barrier methods: A unified framework

It is established that many optimization problems may be formulated in terms of minimizing a function x --> f(0)(x) + H-infinity(f(1)(x), f(2)(x),..., f(m)(x)) + L-infinity(Ax - b), where the f(i) are closed functions defined on R-N, and where H-infinity and L-infinity are the recession functions of closed, proper, convex functions H and L. A is a linear transformation from R-N to a finite dimensional vector space Y with b is an element of Y. A generic algorithm, based on the properties of recession functions, is proposed. This algorithm not only encompasses almost all penalty and barrier methods in nonlinear programming and in semidefinite programming, but also generates new types of methods. Primal and dual convergence theorems are given.