Next: The test example Up: First and second-order optimality Previous: First order necessary conditions

## Second order sufficient optimality conditions

Let be given such that the system of first order necessary conditions is satisfied, i.e. the relations (1.2-1.4), (2.2-2.4) and are fulfilled. Now we state second order conditions, which imply local optimality of . For this purpose, we need the second order derivative of with respect to ,

 (2.6)

Let us assume as in the example below that the state-constraint (1.4) is active at and . Then we require the following second-order sufficient optimality condition:

(SSC)         There exist positive and such that

 (2.7)

holds for all such that
 (2.8)

and
 (2.9) (2.10) (2.11) (2.12)

It is known that (SSC) implies local optimality of in a neighborhood of , see [4]. In our example, we shall verify a slightly stronger condition. We require (2.7) for all , which satisfy only (2.8-2.9).

Next: The test example Up: First and second-order optimality Previous: First order necessary conditions
Hans D. Mittelmann
2003-01-25