Exact Penalty Function And Penalty Algorithm

A typical approach to solve nonlinear programming problems is to augment objective function or the corresponding Lagrangian function using penalty or barrier terms to take account of the constraints (see, e.g., [6,22]). The result-ing merit functions are optimized by utilizing either standard unconstrained (or bound constrained) optimization software or sequential quadratic programming (SQP) techniques, etc. Independently of the technique used, the merit function always depends on a small parameterε(or a large parameterρ=ε-1); for ex-ample, minimizers of the merit function converge to the set of minimizers of the original problem, provided thatεapproaches to 0(?) Particularly, in some SQP approaches, one uses exact penalty functions to produce exact optimizers when e is sufficiently small. However, it should be mentioned that these exact penalty functions have a disadvantage that the evaluation of the merit function either needs Jacobian (see [13,18]) or is no longer smooth (l1 or l∞penalty functions; see [1,2,4,7,10,11,15,16,23,26]). In addition, both kinds of penalty functions may be unbounded below even when the constrained problem is bounded, which may make it difficult or impossible to locate a minimizer.In this paper,we study penalty function method for solving nonlinear constrained programming prob-lems. For nonlinear programming problems, we propose a new class of smooth ex-act penalty functions, which includes both barrier-type and exterior-type penalty functions as special cases. Under mild assumptions, we develop necessary and sufficient conditions for exact penalty property, whose relation to the classical simple exact penalty functions is are also discussed. Based on these, we present a globally convergent penalty algorithm. Finally, promising numerical results on generalized semi-infinite programming problems are reported.