Mathematics for Economists

Constrained local su¢ cient conditions

LetF be the vector function of the constraints and let

C$fxjF(x)= 0 g.It is su¢ cient to show that there is a functionK

such thatKhas a local minimum onCatx 0 withUandK(x 0 )=φ 0 (x 0 )

andK(x)φ 0 (x),x 2 U. In this case ifx 2 U\Cthen

φ 0 (x)K(x)K(x 0 )=φ 0 (x 0 )

sox 0 is a local minimum ofφ 0 onC. Let

K(x)$φ 0 (x 0 )hλ,F(x)iγkF(x)k^2

whereγis a large enough constant. Obviously ifx 2 CthenF(x)= 0

thereforeK(x 0 )=φ 0 (x 0 )asx 02 C.