2.3 Multivariable Optimization with No Constraints 73
The determinants of the square submatrices ofJare
J 1 =
∣
∣k 2 +k 3
∣
∣=k 2 +k 3 > 0
J 2 =
∣
∣
∣
∣
∣
k 2 +k 3 −k 3
−k 3 k 1 +k 3
∣
∣
∣
∣
∣
=k 1 k 2 +k 1 k 3 +k 2 k 3 > 0
since the spring constants are always positive. Thus the matrixJis positive definite
and hence(x 1 ∗, x 2 ∗) orresponds to the minimum of potential energy.c
2.3.1 Semidefinite Case
We now consider the problem of determining the sufficient conditions for the case
when the Hessian matrix of the given function is semidefinite. In the case of a func-
tion of a single variable, the problem of determining the sufficient conditions for
the case when the second derivative is zero was resolved quite easily. We simply
investigated the higher-order derivatives in the Taylor’s series expansion. A simi-
lar procedure can be followed for functions of nvariables. However, the algebra
becomes quite involved, and hence we rarely investigate the stationary points for suf-
ficiency in actual practice. The following theorem, analogous to Theorem 2.2, gives
the sufficiency conditions for the extreme points of a function of several variables.
Theorem 2.5 Let the partial derivatives off of all orders up to the orderk≥2 be
continuous in the neighborhood of a stationary pointX∗, and
drf|X=X∗= 0 , 1 ≤r≤k− 1
dkf|X=X∗= 0
sothatdkf|X=X∗is the first nonvanishing higher-order differential off atX∗. If kis
even, then (i)X∗is a relative minimum ifdkf|X=X∗is positive definite, (ii)X∗is a
relative maximum ifdkf|X=X∗is negative definite, and (iii) ifdkf|X=X∗is semidefinite
(but not definite), no general conclusion can be drawn. On the other hand, ifkis odd,
X∗is not an extreme point off(X).
Proof: A proof similar to that of Theorem 2.2 can be found in Ref. [2.5].
2.3.2 Saddle Point
In the case of a function of two variables,f (x, y), the Hessian matrix may be neither
positive nor negative definite at a point (x∗,y∗) at which
∂f
∂x
=
∂f
∂y
= 0
In such a case, the point (x∗,y∗) is called asaddle point. The characteristic of a
saddle point is that it corresponds to a relative minimum or maximum off (x, y)with
respect to one variable, say,x(the other variable being fixed aty=y∗) and a relative
maximum or minimum off (x, y)with respect to the second variabley (the other
variable being fixed atx∗).
