Second Order Necessary Conditions in Optimization
Jack Warga · SIAM Journal on Control and Optimization · 1984
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. We present here a relatively simple derivation of new second order necessary conditions 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. These new conditions generalize in some respects previous results of Dennis S. Bernstein (A systematic approach to higher-order necessary conditions in optimization theory, SIAM J. Control Optim., 22 (1984), pp. 211–238).