9. Exact Regularization, and Its Connections to Normal Cone Identity and Weak Sharp Minima in Nonlinear Programming

The regularization of a nonlinear program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. In Deng (Pac J Optim 8(1):27–32, 2012), we show that, for a given nonlinear program, the regularization is exact if and only if the Lagrangian function of a certain selection problem has a saddle point, and the regularization parameter threshold is inversely related to the Lagrange multiplier associated with the saddle point. The results in Deng (Pac J Optim 8(1):27–32, 2012) not only provide a fresh perspective on exact regularization but also extend the main results in Friedlander and Tseng (SIAM J Optim 18:1326–1350, 2007) on a characterization of exact regularization of a convex program to that of a nonlinear (not necessarily convex) program. In this paper, we will examine inner-connections among exact regularization, normal cone identity, and the existence of a weak sharp minimum for certain associated nonlinear programs. Along the way, we illustrate by examples, how to obtain both new results and reproduce many existing results from a fresh perspective.