Mathematics for Economists

Mathematics for Economists mainly optimization Medvegyev PÈter 2013 ...

Unconstrained local necessary conditions In the unconstrained casef is deÖned on an open set. Use the "variations"g(t)$f(x 0 +th ...

Unconstrained local su¢ cient conditions Theorem If f is twice di§erentiable at x 0 and fx^0 (x 0 )= 0 and the second derivative ...

Unconstrained local su¢ cient conditions Lemma If xTHx is a positive deÖnite quadratic form onRnthen there is anα> 0 such tha ...

Unconstrained local su¢ cient conditions Asf is twice di§erentiable atx 0 f(x 0 +h)f(x 0 )= fx^0 (x 0 ),h +^1 2 hTfxx^00 (x 0 )h ...

Constrained local necessary conditions Theorem Assume thatφkk= 0 , 1 , 2 ,.. .,p are di§erentiable and assume thatφ 0 has a loca ...

Constrained local necessary conditions Theorem Assume that we are in the regular case andφk,k= 0 , 1 ,.. .,p are twice di§erenti ...

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. ...

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 ...

Constrained local su¢ cient conditions One must show that φ 0 (x)K(x)=φ 0 (x 0 )hλ,F(x)iγkF(x)k^2 That is 0  φ 0 (x 0 )(φ 0 (x ...

Constrained local su¢ cient conditions Obviously (^) F^0 (x 0 )(xx 0 )+o(kxx 0 k) (^2) (^) F^0 (x 0 )(xx 0 ) (^2) +ko(kxx 0 k)k ...

Constrained local su¢ cient conditions By the conditions that F^0 (x 0 )h= 0 )hTL^00 xx(x 0 ,λ)h> 0 ifγis large enough andh 6 ...

Constrained positive deÖnite matrixes Problem LetAbe symmetric. When does xTAx 0 for everyBx= 0 ?Or when does xTAx> 0 for ev ...

Constrained positive deÖnite matrixes One can use the JacobiñSylvester criteria onC. To solve the su¢ ciency problem one should ...

Constrained positive deÖnite matrixes After some calculation one gets that: (^1) Write the basis vectors ofBÖrst.B=(B 1 ,B 2 )=( ...

Constrained positive deÖnite matrixes Example Solvex 12 +x 22 !min,x 1 +x 2 = 1. The Lagrangian isL(x 1 ,x 2 ,λ)=x 12 +x 22 +λ(x ...

Constrained positive deÖnite matrixes Example Solvex 12 +x 22 !min,x 1 +x 2 = 1 , 2 x 1 + 3 x 2 = 4. The only solution is x 1 = ...

Constrained positive deÖnite matrixes Example Solvex 1 +x 2 !min,x 12 +x 22 = 1. The Lagrangian is L(x 1 ,x 2 ,λ)=x 1 +x 2 +λ x ...

Constrained positive deÖnite matrixes Ifx 1 =x 2 = 1 / p 2 thenλ= 1 / p 2 A=  p 2 0 0 p 2  ,B=  2 / p 2 , 2 / p 2  ,H= 0 @ ...

Constrained positive deÖnite matrixes Ifx 1 =x 2 = 1 / p 2 thenλ= 1 / p 2 A=  p 2 0 0 p 2  ,B=  p 2 , p 2  ,H= 0 @ 0 p 2 ...