5.2 Constrained Nonlinear Optimization

5.2 Constrained nonlinear optimization

From a mathematical point of view, we have to consider the following minimization task: Minimize f (x )

;

x = x1 L x m

Subject to n equality constraints: h1(x ) = 0

M hn (x ) = 0

Subject to p inequality constraints: g1(x ) ≤ 0

M gp (x ) ≤ 0

5.2.1

5.2 Constrained nonlinear optimization

5.2.2

Introduction of slack variables x m+1...x m+p to transform the inequality constraints into equality constraints g1( x ) + x m2 +1 = 0

M gp ( x ) + x m2 +p = 0

hn+1( x, x m+1 ) = 0

Written in a more general way

M hn+p ( x, x m+p ) = 0

Introduction of the Lagrange function L L( x, ) = f( x) + 1h1( x) + ... +

h ( x) +

n n

h ( x, x m+1) + ... +

n +1 n +1

Necessary conditions for an extremum are: ∂L =0 ∂x

∂L =0 ∂x1

;

x = x1...x m+p ⇒

M ∂L =0 ∂x m+p

h ( x, x m+p )

n +p n +p

5.2 Constrained nonlinear optimization ∂L =0 ∂ 1

and ∂L =0 ∂

5.2.3

;

= 1...

n +p

M ∂L

⇒

∂

The condition

=0

n+p

∂L = 0 applied on the Lagrange function results in ∂x

∂L ∂x1 ∂L = M = ∇L = 0 ∂x ∂L ∂x m+p

(I)

5.2 Constrained nonlinear optimization

The condition

5.2.4

∂L = 0 is identical with the (n + p) constraints ∂

h1( x) = 0

(II)

M hn+p( x) = 0

The necessary conditions leading to equations (I) and (II) can be written in a more compact form as x1 M x m +p z = F( z) = 0 with (III) 1

M n +p

5.2 Constrained nonlinear optimization

5.2.5

Solving equation (III) by an iterative Newton procedure F( z°) +

∂F z + ... = 0 ∂ z z°

Neglecting higher order terms, we obtain ∂F z = −F( z°) ∂ z z°

Iterative procedure for x and the Lagrange multipliers (i)

∇ 2L

G

(i)

(i)

∆x

∇L

(IV)

= GT

0

∆λ

:

h

5.2 Constrained nonlinear optimization

(i)

(i+1)

= λ

(i)

∆x

x

x

5.2.6

+

(V)

λ

∆λ

∇ 2L is the Hessian matrix of the Lagrange function L , G is the functional matrix connected with the constraints h( x ) :

2

∇ L=

∂∇L ∂∇L K ∂x1 ∂x m+p

;

G=

∂h1 ∂hn+p L ∂x1 ∂x1 M M ∂ h ∂h1 L n+ p ∂x m+p ∂x m+p