A unifying theory of exactness of linear penalty functions II: parametric penalty functions

Abstract

In this article we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

“…If g is Gâteaux differentiable at x , then , where g ′ ( x ) is the Gâteaux derivative of g at x . Finally, if g is merely directionally differentiable at x , then The following theorem, which is a particular case of theorem 3.6 in Reference, contains simple sufficient conditions for the global exactness of the penalty function Φ λ ( x ). For any δ >0, define Ω δ ={ x ∈ A | φ ( x )< δ }.…”

“…Let us note that the rate of steepest descent of the function g at x is closely connected to the so‐called strong slope |∇ g |( x ) of g at x . See References . for some calculus rules for strong slope/rate of steepest descent and the ways one can estimate them in various particular cases.…”

“…Remark In the general case, under the assumptions of Theorem , nothing can be said about locally optimal solutions of the penalised problem /inf‐stationary points of Φ λ on A that do not belong to the set S λ ( c ). In order to ensure that the penalty function Φ λ is completely exact on A (ie, when c =+ ∞ ), one must suppose that the objective function is globally Lipschitz continuous, and there exists a >0 such that for all x ∈ A \Ω (see section 3.3 in Reference).…”

“…Remark Let us note that the assumptions of Theorem cannot be improved (see theorem 3.13 in Reference). On the other hand, the global exactness of the penalty function Φ λ can be proved under weaker assumptions on the penalty term φ .…”

“…In particular, one can apply such popular and efficient modern methods of nonsmooth optimisation as bundle methods, gradient sampling methods, nonsmooth quasi‐Newton methods, discrete gradient method (see also the works of Karmitsa et al), etc. Alternatively, one can utilise smoothing approximations of nonsmooth penalty functions as in References . or the smooth penalty function proposed by Huyer and Neumaier .…”