2ND ORDER NECESSARY CONDITIONS IN OPTIMIZATION

It is known that if a restricted minimization problem satisfies first order necessary conditions for minimum at some point with multiple choices of Lagrange multiplier vectors (or linear functionals) then, in general, second order conditions for different critical variations may require different Lagrange multipliers. A relatively simple derivation of new second order necessary conditions is presented in which different critical variations share a common Lagrange multiplier if they are 'pairwise critical'. The problems that we consider contain restrictions in the form of finitely many equalities and of (possibly infinite-dimensional) inclusions involving arbitrary convex bodies.