Abstract
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 C 2 assumptions. Such an analysis for C 1,1 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.
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© 2003 Springer Science+Business Media Dordrecht
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Rubinov, A., Yang, X. (2003). Optimality conditions. In: Lagrange-type Functions in Constrained Non-Convex Optimization. Applied Optimization, vol 85. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9172-0_6
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DOI: https://doi.org/10.1007/978-1-4419-9172-0_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-4821-4
Online ISBN: 978-1-4419-9172-0
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