Optimality conditions

  • Alexander Rubinov
  • Xiaoqi Yang
Part of the Applied Optimization book series (APOP, volume 85)


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.


Optimality Condition Limit Point Constrain Optimization Problem Unique Minimum Unconstrained Optimization Problem 
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Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Alexander Rubinov
    • 1
  • Xiaoqi Yang
    • 2
  1. 1.School of Information Technology and Mathematical SciencesUniversity of BallaratVictoriaAustralia
  2. 2.Department of Applied MathematicsHong Kong Polytechnic UniversityHong KongChina

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