Engineering Optimization: Theory and Practice, Fourth Edition

108 Classical Optimization Techniques

2.11 If a crank is at an angleθfrom dead center withθ=ωt, whereωis the angular velocity

andtis time, the distance of the piston from the end of its stroke(x)is given by

x=r( 1 −cosθ )+

r^2

4 l

( 1 −cos 2θ )

whereris the length of the crank andlis the length of the connecting rod. Forr= 1

andl=5, find (a) the angular position of the crank at which the piston moves with

maximum velocity, and (b) the distance of the piston from the end of its stroke at that

instant.

Determine whether each of the matrices in Problems 2.12–2.14 is positive definite, negative

definite, or indefinite by finding its eigenvalues.

2.12 [A]=





3 1 − 1

1 3 − 1

− 1 −1 5





2.13 [B]=





4 2 − 4

2 4 − 2

− 4 −2 4





2.14 [C]=





− 1 − 1 − 1

− 1 − 2 − 2

− 1 − 2 − 3





Determine whether each of the matrices in Problems 2.15–2.17 is positive definite, negative

definite, or indefinite by evaluating the signs of its submatrices.

2.15 [A]=





3 1 − 1

1 3 − 1

− 1 −1 5





2.16 [B]=





4 2 − 4

2 4 − 2

− 4 −2 4





2.17 [C]=





− 1 − 1 − 1

− 1 − 2 − 2

− 1 − 2 − 3





2.18 Express the function

f (x 1 , x 2 , x 3 )= −x^21 −x 22 + 2 x 1 x 2 −x 32 + 6 x 1 x 3 + 4 x 1 − 5 x 3 + 2

in matrix form as

f (X)=^12 XT[A]X+BTX+C

and determine whether the matrix [A] is positive definite, negative definite, or indefinite.

2.19 Determine whether the following matrix is positive or negative definite:

[A]=





4 −3 0

−3 0 4

0 4 2



