2.5 Multivariable Optimization with Inequality Constraints viii Contents
where
f
1
0
− 2
=e−^2df
1
0
− 2
=h 1 ∂f
∂x 1
1
0
− 2
+h 2 ∂f
∂x 2
1
0
− 2
+h 3 ∂f
∂x 3
1
0
− 2
=[h 1 ex^3 +h 2 ( 2 x 2 x 3 )+h 3 x 22 +h 3 x 1 ex^3 ]
1
0
− 2
=h 1 e−^2 +h 3 e−^2d^2 f
1
0
− 2
=
∑^3
i= 1∑^3
j= 1hihj∂^2 f
∂xi∂xj
1
0
− 2
=
(
h^21∂^2 f
∂x 12+h^22∂^2 f
∂x^22+h^23∂^2 f
∂x 32+ 2 h 1 h 2∂^2 f
∂x 1 ∂x 2+ 2 h 2 h 3∂^2 f
∂x 2 ∂x 3+ 2 h 1 h 3∂^2 f
∂x 1 ∂x 3)
1
0
− 2
=[h^21 ( 0 )+h^22 ( 2 x 3 )+h^23 (x 1 ex^3 )+ 2 h 1 h 2 ( 0 )+ 2 h 2 h 3 ( 2 x 2 )+ 2 h 1 h 3 (ex^3 )]
1
0
− 2
=− 4 h^22 +e−^2 h^23 + 2 h 1 h 3 e−^2Thus the Taylor’s series approximation is given by
f(X)≃e−^2 +e−^2 (h 1 +h 3 )+1
2!
(− 4 h^22 +e−^2 h^23 + 2 h 1 h 3 e−^2 )whereh 1 =x 1 − , 1 h 2 =x 2 , andh 3 =x 3 +. 2
Theorem 2.3 Necessary Condition Iff(X) has an extreme point (maximum or min-
imum) atX=X∗and if the first partial derivatives off(X)exist atX∗, then
∂f
∂x 1(X∗)=
∂f
∂x 2(X∗) =· · · =
∂f
∂xn(X∗)= 0 (2.9)
Proof: The proof given for Theorem 2.1 can easily be extended to prove the present
theorem. However, we present a different approach to prove this theorem. Suppose that
one of the first partial derivatives, say thekth one, does not vanish atX∗. Then, by
Taylor’s theorem,
f (X∗+ h)=f(X∗)+∑ni= 1hi∂f
∂xi(X∗)+R 1 (X∗,h)