Transcript PowerPoint

Introduction to Multivariate
Optimality
Lecture VI
Introduction to Multivariate
Optimality (I)
 Development
of the Unconstrained
optimum
• The vector form of the Taylor Series Expansion
is
f ( x)  f ( x )  x f ( x )dx  dx'  f ( x )dx
*
*
1
2
2
xx
*
Introduction to Multivariate
Optimality (II)
• By similar arguments as discussed in the
univariate case we can then define
f ( x)  f ( x )   x f ( x)dx  dx'  f ( x ) dx  0
*
1
2
2
xx
*
Introduction to Multivariate
Optimality (III)
 Constrained
Multivariate Optimum
• The general problem of the constrained
multivariate optimum can be defined as
max f ( x )
x
st G ( x )  b
Introduction to Multivariate
Optimality (IV)
• Most of the problem in the constrained
optimum comes in defining a feasible
perturbation
– We start from a feasible point and use a Taylor
expansion of the multivariate constraint
G( x)  G( x )   x G( x ) dx
*
*
Introduction to Multivariate
Optimality (V)
– A critical part of this discussion is the Jacobian
  G1 ( x )

  x1
  G2 ( x )
 x G ( x )   x1


  G ( x)
m

  x1
 G1 ( x )
 G1 ( x ) 


 x2
 xn 
 G2 ( x )
 G2 ( x ) 

 x2
 xn 

 Gm ( x )
 Gm ( x ) 


 x2
 xn 



Introduction to Multivariate
Optimality (VI)
• Given the Taylor series expansion of the vector
equation, we can see that starting from a
feasible and stepping to another feasible point
involves solving the equation
G( x )  G( x )   x G( x ) dx  0
Introduction to Multivariate
Optimality (VI)
• Using this information to solve for a
perturbation which maintains feasibility
requires first splitting the dx vector up into an
m*m portion and a m*(n-m) portion
G
1
 d x1 
G2 
0

d x2 

G1d x1  G2 d x2  0
Introduction to Multivariate
Optimality (VII)
• Solving for the feasible change
1
1
d x1   G G2 d x2
Introduction to Multivariate
Optimality (VIII)
• Returning to the original unconstrained
objective function and using our familiar Taylor
series expansion
f ( x)  f  x *    x f ( x * ) d x  12 d x' 2x f ( x * ) d x  0
f
1
 d x1  1
f2 
 2 d x1 d x2

d x2 



 F11
F
 21
F12   d x1 
0



F22  d x2 
Introduction to Multivariate
Optimality (IX)
• Substituting for the feasible changes in x from
the expansion of the constraint matrix, we have
 f1
 G11 G2 d x2  1 
1
f2  
  2  d x2G2G1 
d x2


 F11


d x2 
  F21
F12   G11 G2 d x2 

0

F22  
d x2

 f1 G11 G2 d x2  f 2 d x2  0
 f 2  f1 G11G2  d x2  0
f 2  f1 G11G2  0
 The
second-order necessary conditions are
then defined by
 d x G G 
2
2

1
1
 F11


d x2 
  F21
 dx G G 1 F  dx  F
2 21
 2 2 1 11
F12  G11G2 dx2 


F22   dx2 
1


G
1
1 G2 dx2




dx2 G2 G1 F12  dx2 F22 
  dx2 
 dx G G 1 F G 1G dx  dx  F G 1G dx  dx G G 1 F dx  dx  F dx 
2 21 1
2
2
2
2 1
12
2
2 22
2
 2 2 1 11 1 2 2

dx2 G2G11 F11G11G2  F21G11G2  G2G11F12  F22  dx2

