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### 1.2.2.2 Second-order conditions

For the sake of completeness, we quickly state the second-order conditions for constrained optimality; they will not be used in the sequel. For the necessary condition, suppose that is a regular point of and a local minimum of over , where is defined by the equality constraints (1.18) as before. We let be the vector of Lagrange multipliers provided by the first-order necessary condition, and define the augmented cost as in (1.27). We also assume that is . Consider the Hessian of with respect to evaluated at :

The second-order necessary condition says that this Hessian matrix must be positive semidefinite on the tangent space to at , i.e., we must have for all . Note that this is weaker than asking the above Hessian matrix to be positive semidefinite in the usual sense (on the entire ).

The second-order sufficient condition says that a point is a strict constrained local minimum of if the first-order necessary condition for optimality (1.25) holds and, in addition, we have

 such that (1.29)

Again, here is the vector of Lagrange multipliers and is the corresponding augmented cost. Note that regularity of is not needed for this sufficient condition to be true. If is in fact a regular point, then we know (from our derivation of the first-order necessary condition for constrained optimality) that the condition imposed on in (1.29) describes exactly the tangent vectors to at . In other words, in this case (1.29) is equivalent to saying that is positive definite on the tangent space .

Next: 1.3 Preview of infinite-dimensional Up: 1.2.2 Constrained optimization Previous: 1.2.2.1 First-order necessary condition   Contents   Index
Daniel 2010-12-20