Optimality conditions for an isolated minimum of order two in C1 constrained optimization

In this paper we obtain first and second-order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint. First-order sufficient conditions are derived in terms of generalized convex functions. In the necessary conditions we suppose that the data are continuously differentiable. A notion of strongly KT invex inequality constrained problem is introduced. It is shown that each Kuhn–Tucker point is an isolated global minimizer of order two if and only if the problem is strongly KT invex. The article could be considered as a continuation of [I. Ginchev, V.I. Ivanov, Second-order optimality conditions for problems with C1 data, J. Math. Anal. Appl. 340 (2008) 646–657].