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.

arXiv commentThis is a slightly edited version of Accepted Manuscript of an article published by Taylor & Francis in Optimization on 06/07/2017

References in corpus (4)

• Error Bounds and Metric Subregularity
• A Unifying Theory of Exactness of Linear Penalty Functions
• Smooth Exact Penalty Functions: A General Approach
• Smooth exact penalty functions II: a reduction to standard exact penalty functions

Cited by in corpus (7)

• Exact Penalty Functions for Optimal Control Problems I: Main Theorem and Free-Endpoint Problems
• Exact penalty functions with multidimensional penalty parameter and adaptive penalty updates
• Exact Penalty Functions for Optimal Control Problems II: Exact Penalisation of Terminal and Pointwise State Constraints
• Augmented Lagrangian Functions for Cone Constrained Optimization: the Existence of Global Saddle Points and Exact Penalty Property
• A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness
• New global optimality conditions for nonsmooth DC optimization problems
• A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness