Optimality conditions

Nonsmooth analysis will play an important role in this chapter. Various calculus rules, such as, mean-value theorem, chain rule and Taylor expansion have been established, see [13, 24, 26, 23, 102, 125, 128]. In this chapter, we consider the convergence of first-order necessary condition and second-order necessary condition that are obtained by Lagrange-type and augmented Lagrangian problems to that of constrained optimization problems. In the literature, various methods have been investigated. Arc methods and penalty methods were given by [87] and [5] for inequality constrained optimization problems under C2 assumptions. Such an analysis for C1,1 optimization problems has been given in [126]. A method that combines curvilinear paths and trust regions is given in [19] for a unconstrained optimization problem.