Strong KKT, Second Order Conditions and Non-solid Cones in Vector Optimization

  • Joydeep Dutta
Part of the Vector Optimization book series (VECTOROPT, volume 1)


In this chapter we shall concentrate on studying the Karush–Kuhn–Tucker (KKT) type optimality conditions for both Pareto and weak Pareto minimum of a usual vector optimization problem, that is, a vector optimization problem with equality and inequality constraints. It is now a well known fact that the KKT conditions for scalar optimization problems play a major role in the analysis of algorithms. It is still not clear whether the KKT conditions for a vector optimization problem plays such a significant role. However that does not mean that there is really no use studying them. They do play a fundamental role in understanding the nature of the solutions of a vector optimization. Further they can always be used as an optimality certificate through which we can conclusively decide that a point is not a Pareto minimum or a weak Pareto minimum. Optimality conditions can also lead to the design of certain merit functions which can lead to robust error bounds for a convex vector optimization problems with strongly convex objective functions. Thus it is important for us to develop necessary and sufficient optimality conditions for a vector optimization problem.


Vector Optimization Tangent Cone Vector Optimization Problem Order Optimality Condition Asplund Space 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKanpurIndia

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