An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems

A group of curves generates a new curve which is called an envelope. When one deals with a minimization problem with infinitely many inequality constraints, one must encounter an envelopelike effect caused by the constraints. In this paper we present second-order necessary conditions, which involve a new term besides the second derivative of the Lagrange function. We apply our results to minimizing problems of sup-type functions. One will observe in examples that the new term given in this paper explains well the behavior of the second directional derivative of the sup-type function.