Transposition theorems and qualification-free optimality conditions

Hermann Schichl (hermanesi.ac.at)
Arnold Neumaier (Arnold.Neumaierunivie.ac.at)

Abstract: New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two Karush-John optimality conditions -- holding without any constraint qualification -- are proved for single- or multi-objective constrained optimization problems. The first condition applies to polynomial optimization problems only, and gives for the first time necessary and sufficient global optimality conditions for polynomial problems. The second condition applies to smooth local optimization problems and strengthens known local conditions. If some linear or concave constraints are present, the new version reduces the number of constraints for which a constraint qualification is needed to get the Kuhn-Tucker conditions.

Keywords: certificate of global optimality, first order optimality conditions, Fritz John conditions, Karush-John conditions, global optimality condition, global optimization, Kuhn-Tucker conditions, Mangasarian-Fromovitz constraint qualification, necessary and sufficient conditions, Positivstellensatz, second order optimality conditions, theorem of the alternative, transposition theorem

Category 1: Global Optimization (Theory )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Other Topics (Multi-Criteria Optimization )

Citation: submitted, May 11, 2005.

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Entry Submitted: 05/11/2005
Entry Accepted: 05/12/2005
Entry Last Modified: 02/20/2006

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