Legendre-type optimality conditions for a variational problem with inequality state constraints

, they differ from the Legendre-Clebsch condition. They give information about the Hesse matrix of the integrand at not only inactive points but also active points. On the other hand, since the inequality state constraints can be regarded as an infinite number of inequality constraints, they sometimes form an envelope. According to a general theory [9], one has to take the envelope into consideration when one consider second-order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint. However, we show that we do not need to take it into account when we consider Legendre-type conditions. Finally, we show that any inequality state constraint forms envelopes except two trivial cases. We prove it by presenting an envelope in a visible form.

Authors and Affiliations

1. Faculty of Economics, Toyama University, Gofuku 3190, Toyama 930-8555, Japan, e-mail: koga@eco.toyama-u.ac.jp, , , , , , JP

   Sayuri Koga

2. Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan, e-mail: kawasaki@math.kyushu-u.ac.jp, , , , , , JP

   Hidefumi Kawasaki

Received April 18, 1995 / Revised version received January 5, 1998 Published online August 18, 1998

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