Optimization

60 Optimization: Algorithms and Applications

Newton, and Levenberg–Marquardt methods. Consider two springs of unit

length and with stiffness k

1

and k

2

, joined at the origin. The other two ends

of the springs are xed on a wall (see Figure 3.5). On applying a force, the

spring system will deect to an equilibrium position, which we are inter-

ested in determining. The potential of the spring system is given by

U k x x k x x F x

x

= + + −

( )

+ + − −

( )

− +

1 1

2

2

2

2

2 1

2

2

2

2

1

1 1 1 1

1

( ) ( ) ( FF x

x

2

2

) (3.7)

where

F F

x x

1 2

,

( )

is the force applied at the origin due to which it moves to a posi-

tion (x

1

, x

2

). Assuming k

1

= 100 N/m, k

2

= 90 N/m, and

F F