Principles of Mathematics in Operations Research

5.1 Minima, Maxima, Saddle points 73

Theorem 5.1.3 The extreme values for f(x,y) can occur only at

i. Boundary points of the domain of f.

ii. Critical points (interior points where fx = fy = 0, or points where fx or

fy fails to exist).

If the first and second order partial derivatives of f are continuous throughout

an open region containing a point (a,b) and fx(a,b) = fy(a,b) — 0, you may

be able to classify (a, b) with the second derivative test:

i- fxx < 0, fxxfyy - fL > 0 at (a,b)

ii- fxx > 0, fxxfyy ~ /Jy > 0 at (a,b)

in- fxxfyy - fc2^2 y<0 at (a,b)

iv- fxxfyy- fL = 0 at (a,b)

local maximum;

local minimum;

saddle point;

test is inconclusive (f is singular).

5.1.2 Quadratic forms

Definition 5.1.4 The quadratic term f(x,y) = ax^2 + 2bxy + cy^2 is positive

definite (negative definite) if and only if a > 0 (a < 0) and ac—b^2 >0.f has

a minimum (maximum) at x = y = 0 if and only if fxx(0,0) > 0 (/Xx(0,0) <

0) and fxx(0,0)fyy{0,0) > f^2 y{0,0). If f(0,0) = 0, we term f as positive

(negative) semi-definite provided the above conditions hold.

Now, we are able to introduce matrices to the quadratic forms:

ax^2 + 2bxy + cy^2 — [x, y)

a b

be

Thus, for any symmetric A, the product / = x^7 Ax is a pure quadratic

form: it has a stationary point at the origin and no higher terms.

xATx - [xi,x 2 ,--- ,xn]

an

021

an\

aX2 •

"22 •

0-n2 •

• 0-in


• «2n

ann _

Xi

X2

%n

= anZj +a 12 xix 2 H V annx^2 n = ^ ^ ai}-XjXj.

i=l j=l

Definition 5.1.5 If A is such that a^- = dxJx. (hence symmetric), it is
called the Hessian matrix. If A is positive definite (xTAx > 0, Vx ^ 6) and
if f has a stationary point at the origin (all first derivatives at the origin are
zero), then f has a minimum.