Transcript AOE/ESM 4084 Engineering Design Optimization Lecture # 11 Indirect or Transformation Methods

AOE/ESM 4084
Engineering Design Optimization
Lecture # 11
Indirect or Transformation Methods
Zafer Gürdal
Virginia Tech
5/24/2016
•Indirect or Transformation Methods(or
SUMT Techniques) for N-Dimensional
Constrained Minimization
• Penalty Function Approaches
– Exterior penalty Function Method
– Interior Penalty Function Method(Barrier Function)
– Extended Interior Penalty Function Method
• Method of Multipliers(Augmented Lagrange Multiplier Method)
• Those methods which take into account the constraints explicitly
and search the design space while computing the objective
function and constraints to guide how to compute the search
direction and decide how far to move in a given direction by
checking constraints are referred to as the direct methods or
direct search methods.
•Penalty Function Approaches
• Standard Mathematical Statement
– Minimize
f ( x)  f ( x1 , x2 ,..., xn )
– Subject to g j ( x )  0
j  1,..., ng
hk ( x )  0
k  1,..., n e
• Pseudo-objective Function
– Minimize
 ( x, rp )  f ( x )  rp P( x )
where scalar rp is the penalty multiplier and P ( x ) is the
penalty function which depends on the type of constraint
(equality v.s. inequality) and the penalty method used (
Exterior, Interior, Extended Interior)
•Exterior Penalty Function Approach
• Penalty Function
ne
ng
i 1
j 1
P ( x )   [hk ( x )]2  {max[ 0, g j ( x )]}2
• When all constraints are satisfied, P( x )  0 .
• Starting from a feasible design point, minimization of the
pseudo-objective function will immediately take the design
into the infeasible design space (the optimizer will not know
any constraints).
• If the starting point is in the infeasible design space, the large
the penalty multiplier is, the closer the minimum of the pseud
function to the constrained min.
• If the penalty multiplier is large, then obtaining the
unconstrained minimization is difficult due to ill conditioning
of the pseudo-objective function.
x
)  0.5 x
r gnisaercni
g ( x)  2 xgo
p
 (r , x )
f (x )
•Interior penalty Function Approach
• Penalty Function
– Different versions exist,
n
1
P( x )  
( x )is a convenient one.
j 1 g j ( x )
• The pseudo-objective function is defined only in the
feasible region. Must have a feasible initial starting point.
g
• The smaller the penalty multiplier is, closer is the
minimum of the pseudo-objective function to the
minimum of the constrained problem.
• The pseudo-objective function is discontinuous at the
constraint boundaries, extreme caution must be used in
one dimensional search to ensure that a feasible
minimum is obtained.
• Equality constraints treated same as Exterior method
with
penalty.
1 / rp
x
x )  0.5 x
p
r gnisaerced
 (r , x )
g ( x)  2 xgo
f (x )
•Extended Interior Penalty Function Approach
1
Pj ( x ) 
g j (x)
go
• Penalty function defined differently in the different
regions of the design space with a transition point,
Quadratic penalty.
if g j (x)  go
2


g
(
x
)

 g j ( x )
 1  j

Pj ( x ) 

3

3



 g 
g
g j ( x ) 
o
o






• No discontinuity at the constraint boundaries.
• Either feasible or infeasible starting point.
• Method operates in the feasible design space.
if g j (x)  go
• Transition from interior to exterior governed by
go  C ( rp )
a
1
1
a
3
2
where
and
C  0.2
• Other Extended Interior Penalty Functions
– Linear Extended Penalty Function
1
Pj ( x ) 
g j (x)
Pj ( x )  
if g j (x)  go
2 go  g j ( x )
g
2
o
– Variable Penalty Function
if g j (x)  go
x
f ( x )  0.5 x
p
r gnisaerced
go
 (r , x )
g ( x)  2 xgo
f (x )