Optimality conditions in mathematical programming and composite optimization

New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat’s rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki and Cominetti; but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. The paper points out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. The conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral. The paper recovers the classical conditions of Ben-Tal-Zowe and Cominetti.