In this note we give an elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets.

Additional Metadata
Keywords Fritz-John conditions, Karush–Kuhn–Tucker conditions, nonlinear programming
JEL Optimization Techniques; Programming Models; Dynamic Analysis (jel C61), Information and Product Quality; Standardization and Compatibility (jel L15), Business Administration and Business Economics; Marketing; Accounting (jel M), Management of Technological Innovation and R&D (jel O32)
Persistent URL dx.doi.org/10.1016/j.ejor.2006.04.012, hdl.handle.net/1765/19255
Series ERIM Top-Core Articles , Econometric Institute Reprint Series
Journal European Journal of Operational Research
Citation
Birbil, S.I, Frenk, J.B.G, & Still, G.J. (2007). An elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions in nonlinear programming. European Journal of Operational Research, 180(1), 479–484. doi:10.1016/j.ejor.2006.04.012