Abstract. This paper contains several remarks and results on the necessary optimality conditions of Fritz John type for a nonlinear programming problem with inequality and equality constraints. The first part is concerned with a simple and brief proof of the Fritz John conditions, proof which benefits from the use of the Bouligand tangent cone. The second part is concerned with the same problem, where a set constraint (which covers the constraints which cannot be expressed by means of neither equalities nor inequalities) is added. Several remarks and results are given to deepen the study of Fritz John type conditions for this case.

Received: May 17, 2011

AMS Subject Classification: 90C30, 90C46

Key Words and Phrases: Fritz John conditions, set constraint

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