Computer Aided Engineering Design

OPTIMIZATION 353

B

y A

2

1

0

–1

–2

–2 –1 0 1 2 3

x

Figure 12.9 Function (thin lines) and constraint (thick line)

curves for Example 12.7

Example 12.8. Find the optimal points for f(x,y) = 2 + xy on the circle (x – 2)^2 + (y – 2)^2 = 4.
The Lagrangian becomes
L = 2 + xy + λ[(x – 2)^2 + (y– 2)^2 – 4]

differentiating which yields

∂L/∂x=y + 2λ(x – 2) = 0

∂L/∂y = x + 2λ(y – 2) = 0

solving which we get y = x. Using the constraint, we have

2(y – 2)^2 = 4

ory = 2 ±√2. Thus, the two solution sets are (2 + √2, 2 + √2) and (2 – √2, 2 – √2). The following
summarized solution table is provided along with the function values.

xy λ f(x,y)

(i) 2 + √22 + √ 2 – –^1212 13.66

(ii) 2 – √22 – √ 2 12 –^12 2.34(

The Hessian is computed as

H =

21

12

λ

λ

⎡

⎣

⎢

⎤

⎦

⎥

It can be shown that the Hessian for solution (i) is negative definite and for (ii) it is neither positive
nor negative definite. Thus, solution (i) (point C in Figure 12.10) provides a local maximum while
(ii) (point B) provides a saddle point.

12.3.3 Multivariable Optimization with Inequality Constraints

Consider now, minimizing a function f(X) in n variables with m inequality constraints gi(X)≤ 0,