Constrained local necessary conditions

Theorem

Assume thatφkk= 0 , 1 , 2 ,.. .,p are di§erentiable and assume thatφ 0

has a local minimum at x 0 on the set

X$fxjφk(x)= 0 ,k= 1 ,.. .,pg.

Then there are multipliers

l=(λ 0 ,λ 1 ,.. .,λp) 2 Rp+^1

such that p

∑

k= 0

λkφ^0 k(x 0 )= 0.

Ifφ^0 k(x 0 ),k= 1 ,.. .,p are linearly independent thenλ 0 = 1 is possible.