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It is well known that second-order information is a basic tool notably in optimality conditions and numerical algorithms. In this work, we present a generalization of optimality conditions to strongly convex functions of order  γ  with the help of first- and second-order approximations derived from (Optimization 40(3):229-246,  2011 ) and we study their characterization. Further, we give an example of such a function that arises quite naturally in nonlinear analysis and optimization. An extension of Newton’s method is also given and proved to solve Euler equation with second-order approximation data.

Second-order optimality conditions for nonlinear programs and mathematical programs