chapter 7 handling constraints. nlp with linear constraints: optimize domain constraints:...
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Chapter 7Handling Constraints
NLP with linear constraints:Optimize
Domain constraints:
Equalities:
Inequalities:
for ,iii uxl qi ,,1
bAx ,,, where 1
qq Rxx x ,)( ijaA pbbb ,,1
pi 1,1 qj
Rxxf q ),...,( 1
dCx ,,, where 1 qxx x ,)( ijcC mddd ,,1
mi 1,1 qj
NonLinear Programming Problem
GENOCOP I
Original GENOCOP (GEnetic algorithm, for Numerical Optimization for COnstrained Problems):
With linear constrain An elimination of the equalities (convex) Special “genetic” operators
Example Optimize a function of six variables:
subject to the following constraints
Express four variables as functions of the remaining two:
),,,,,( 654321 xxxxxxf
0,0,0,0,0,0
4
3
10
5
654321
52
41
654
321
xxxxxx
xx
xx
xxx
xxx
216
25
14
213
3
4
3
5
xxx
xx
xx
xxx
Reduce the original problem to the optimization problem of a function of two variables and :
Subject to the following constraints (inequalities only):
These inequalities can be further reduced to:
))3(),4(),3(),5(,,(),( 2121212121 xxxxxxxxfxxg
.03
04
03
05
0,0
21
2
1
21
21
xx
x
x
xx
xx
.5
04
03
21
2
1
xx
x
x
Equality constraint set:
Split A:
New set of inequalities (after removal ):
Split C:
bAX
bXAXA 22
11
22
11
11
1 XAAbAX
equations)equality t independen (,,, where
21
1
pxxxX
piii
12
21
11
11 uXAAbAl
1X
dCX dXCXC 2
21
1
bACdXAACC 111
22
1112 )(
Elimination of Equalities
Final set of constraints:
original domain constraints:
new inequalities:
original inequalities (after removal of variables):
112
211 uAXAblA
bACdXAACC 111
22
1112 )(
222 uXl
1X
Example (1) Optimize a function of six variables:
subject to the following constraintsDomain constraints:
Equalities:
Inequalities:
),,,,,( 654321 xxxxxxf
120
34
103
62
55 ,020 ,515
,010 ,5075 ,4020
52
41
653
321
654
321
xx
xx
xxx
xxx
xxx
xxx
Example (2)
Transportation problem:
1211109
8765
4321
1211108765431 )181614209712112010(
xxxx
xxxx
xxxx
xxxxxxxxxx
5
25
15
1015155
minimize
Representationfloating point representation
Initialization process A subset of potential solutions -- the space of the
whole feasible region (randomly) The remaining subset -- the boundary of the
solution space.
Genetic operators
dynamic non-uniform.
Mutation
Uniform mutation
Boundary mutation
Non-uniform mutation
mk vvvx ,,,1 mk vvvx ,,,, '1
'
],[ )()(' t
vtv s
kskk ulv
},{ )()(' t
vtv s
kskk ulv
vv t u v
v t v lkk k k
k k k
' ( , )
( , )
if a random digit is 0
if a random digit is 1
( , ) ( )( )
t y y rt
Tb
11
Crossover Arithmetical crossover
Simple crossover
Heuristic crossover
)1(
)1(
12'2
21'1
axaxx
axaxx
)()(
:
]1..0[:
)(
12
2123
xfxf
w
r
xxxrx
thanbetter is
number attemp
number random
)1(,),1(,,,
)1(,),1(,,,
111'2
111'1
ayaxayaxyyx
axayaxayxxx
qqkkk
qqkkk
GENOCOP II
With non-linear constrain Distinguish between linear and nonlinear
constraints A single starting point Quadratic penalty function Iterative execution of GENOCOP
Algorithm
Procedure GENOCOP II
begin
split the set of constraints C into
select a starting point ( need not be feasible.)
set the set of active constraints, A to (V: violated constraints
at point )
set penalty
C L N Ne i
xsxs
r r 0
t 0
A N Ve xs
while (not termination-condition) dobegin
execute GENOCOP I for the function
with linear constraints L and the starting point save the best individual :
update A:
decrease penalty r:
(where ; end
end
xs
t t 1
AAxfrxF Tr2
1)(),(
x*
x xs *
VSAA
r g r t ( , )
);1,(),( trgrtrg ,0r ) 1)0,( rg
Example
Minimize
s.t.
yxyxf ),(
369688324:2
2882:1234
234
xxxxyc
xxxyc
IterationNumber
01234
The bestpoint(0,0)(3,4)
(2.06, 3.98)(2.3298, 3.1839)
(2.3295, 3.1790)
ActiveConstraints
nonec2
c1 , c2
c1 , c2
c1 , c2
Other Techniques
Homaifar
Joines and Houck
mjqxh
qjxgxf
xpxfxeval
,m,qjxh
,q,ixg
xf
j
j
j
j
i
1)(
1)}(,0max{)(
)()()(
10)(
10)(
)(
if ,
if ,
s.t.
min
m
j jij xfRxp1
2 )()(
m
j j xftCxp1
)()()(
Schoenauer and Xanthakis1. Start with a random population of individuals
(feasible or infeasible)
2. Set ( j is a constraint counter)
3. Evolve this population with , until a given percentage of population (flip threshold ) is feasible for this constraint
4. Set
5. The current population is the starting point for the next phase of the evolution, where .
6. If , repeat the last two steps, otherwise optimize the objective function ,
1 jj
1j
)()( xfxeval j
)()( xfxeval jmj )( mj
)()( xfxeval
Powell and Skolnick
otherwise
if
m
j jFx
Fx
m
j j
xfrxf
xf
Fx
xt
xtxfrxP
1
1
)}},()({min
)}({max,0max{
,0
),(
),()()(
Bean and Hadj-Alouane
otherwise (t),
allfor if (t),
allfor if (t),)1/
2
1
tiktFiB
tiktFiB
t
XfXPm
j j
1)(
1)((
)1(
)()(1
2
GENOCOP III Two separate populations
(search point): satisfy linear constraints
(reference point): satisfy all constraints Repair:
Feasible points: (reference point )
Infeasible search points: (search point )
( : better reference points)
( is feasible)
( :probability of replacement)
(if is better than )
( :probability of replacement)
Ps
Pr
R
eval R f R( ) ( )
Z aS a R ( )1 Reval S eval Z f Z( ) ( ) ( ) Z
S Z
pr
R Z f Z( ) f R( )
S Z
pr
S
Extend GENOCOP III
Nonlinear equations 0)( Xh j
)(Xh j