Constrained local su¢ cient conditions
Theorem
Assume L^0 x(x 0 ,λ)= 0 ,that is let x 0 be a stationary point of the
Lagrangian. Assume that we are in the regular case and
φk,k= 0 , 1 ,.. .,p are twice di§erentiable at x 0 .If
hTL^00 xx(x 0 ,λ)h> 0 , h 6 = 0
wheneverhφk^0 (x 0 ),hi= 0 ,k= 1 , 2 ,.. .,p then x 0 is a local minimum of
the constrained optimization problem. If
hTL^00 xx(x 0 ,λ)h< 0 , h 6 = 0
wheneverhφk^0 (x 0 ),hi= 0 ,k= 1 , 2 ,.. .,p then x 0 is a local maximum of
the constrained optimization problem.
