1ST AND 2ND-ORDER NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR INFINITE-DIMENSIONAL PROGRAMMING-PROBLEMS

First-order and second-order necessary and sufficient optimality conditions are given for infinite-dimensional programming problems with constraints defined by arbitrary closed convex cones. The necessary conditions are immediate generalizations of those known for the finite-dimensional case. However, this does not hold for the sufficient conditions as illustrated by a counterexample. Here, to go from finite to infinite dimensions, causes an essential change in the proof-techniques and the results. We present modified sufficient conditions of first-order and of second-order which are based on a strengthening of the usual assumptions on the derivative of the objective function and on the second derivative of the Lagrangian. © 1979 The Mathematical Programming Society.