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  and  for inequality constrained optimization problems under C 2 assumptions. Such an analysis for C 1,1 optimization problems has been given in . A method that combines curvilinear paths and trust regions is given in  for a unconstrained optimization problem.
KeywordsOptimality Condition Limit Point Constrain Optimization Problem Unique Minimum Unconstrained Optimization Problem
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