wW,+ A+Wj W,"NwWAW,5 6+
WA,+Vw 5A+WA +BAW Luc Jaulin and Didier Henrion,
LISA-Angers WA 2ff
September 29, 2003
W AWVAW5 Model : +m ETc | ' ReR5|.
Parameters : Rc R2
Sampling times : |c |2c c |6
Data bars : d+c +noc d+2c +2noc c d+6c +6no Feasible set for the parameters
V ' iT 5 U2c +m ETc | 5 d+c +noc
+m ETc |6 5 d+6c +6noj
Presentation of the software 5i|_i4L
As a conclusion,
- interval methods are fast when the number ? of variables is small (here 2).
- when ? is large, constraint propagation can be used to limit the number of bisections
WW +a,A+5
A SJ?r|o@?| D E%c c %? is a subset of U?. A RoJeS|Jo associated with D is a function
FD G d o $ dd o_Do
A constraint D is RoJeS|@K,e if a polynomial projector is available for D.
Presentation of the solver hL2_
Among projectable constraints, we have @S+S,S SJ?r|o@?|r such as
%2 n % t? E%2 f
%2 n i T E%e '
which can be projected using interval arithmetic, ,?e@o SJ?r|o@?|r such as
S% e%2 n S%e n 2 f b% n % n S%e D ' f
which can be projected using linear programming,
,?e@o 6@|o% ?e^@,|er (LMI) such as 3
EC
f 2 f 2 e
4 FDn%
3 EC
b 2 2 2 e D 2 D 2
4
FDn n%? 3 EC
. . e e
4
FD f
which can be projected using convex optimization techniques.
WWW #,5WAW
Decompose the CSP to be solved into a conjunction of projectable constraints.
, @4T*i: The CSP D G
;?
=
+ ' % n 2%2 n %
+2 ' %2 n % s %4
%54n%556
can be decomposed into the following projectable constraints:
D G
+ + ' % n 2%2 n %
+2 ' %2 n % 5
D2 G 5 ' %*52
D G 52 ' s
%2 n %22
WV +BAW Consider the three following constraints
D G + ' %2c D2 G %+ ' c
D G + ' 2% n
To each variable %c +, we give the domain o 4c 4d. J?r|o@?| RoJR@}@|J? projects all constraints upto the equilibrium.
E , + 5 o 4c 4d2 ' dfc 4d E2 , % 5 *dfc 4d ' dfc 4d
E , + 5 dfc 4d _ EE2 dfc 4dn
' dfc 4d _ Eo 4c o ' dfc o
% 5 dfc 4d _ Edfc o*2 n *2 ' dfc *2o
E , + 5 dfc o _ dfc *2o2 ' dfc *eo E2 , % 5 dfc *2o _ *dfc *eo ' >
+ 5 dfc *eo _ *> ' >
V ww 5W5A,v
If FD and FE are two projectors for two D and E, the contractor
FD FE FD FE FD FE is not necessarily a projector for D _ E.
hLM*i4: Decrease the pessimism due to local consistency.
5L*|L?: Increase the class of projectable con- straints to avoid unnecessary decompositions.
VW wW,+ A+Wj W,"NwWAv The constraint E%c %2 given by
3 EC
n e% n .%2 e n D% %2 S n 2% n 2%2
e n D% %2 2 n D% n D%2 b n D% n %2
S n 2% n 2%2 b n D% n %2 n f% n H%2
4
FD f
or equivalently 3
EC
e S e 2 b S b
4
FDn% 3 EC
e D 2 D D D 2 D f
4
FDn%2 3 EC
. 2 D H
4
FD f
is an LMI.
w?i@h UL?t|h@?|t @hi wWt
A set of linear inequalities is an LMI. For instance + @% n @2%2 n K f
@2% n @22%2 n K2 f is equivalent to the following LMI
# @% n @2%2 n K f
f @2% n @22%2 n K2
$
fc
e,
# K f f K2
$
n %
# @ f f @2
$
n %2
# @2 f f @22
$
f
,**TtL_t @hi wW ti|t (Schur complement theorem).
%2 n 2%22 2%%2 D / 3 EC
% %2
% 2
%2 4
FD f
/ 3 EC
f f f 2 f
4
FD n % 3 EC
f f
f f f f f
4
FD n %2 3 EC
f f f f f
f f 4
FD f
wWt @hi ?L*i_ ? }*LM@* LT|43@|L?
Function :
s E ' e%2 %2%e n %S n %2%2 e%22 n %%e2 Gradient
} E '
# S% e%2% n S%D n 2%%2 n %e2
%e n %2%22 H%2 n e%%2
$
Hessian matrix#
H 2%2%2 n f%e n 2%2 e% n S%%22 n e%2 e% n S%%22 n e%2 S%2%2 H n 2%%22
$
L?t|h@?|t
;A AA AA A? AA AA AA
=
e%2 %2%e n %S n %2%2 e%22 n %%e2 sn S% e%%2 n S%D n 2%%2 n %e2 ' f
%e n %2%22 H%2 n e%%2 ' f
# H 2%2%2 n f%e n 2%2 e% n S%%22 n e%2 e% n S%%22 n e%2 S%2%2 H n 2%%22
$
f
#iUL4TLt|L? : The previous set of constraints can be decomposed into the following set of
projectable constraints
D G
+ @ ' %2c @2 ' %c @ ' %ec @e ' %Dc @D ' %Sc K ' %22c K2 ' %2c K ' %e2c
D2 G S ' @%2c S2 ' @K2c Se ' @2%2c SS ' @Kc D G SH ' %Kc S ' %Kc S. ' @%2c SD ' %K2c De G
;A
? A=
e@ S n @D n S2 eK n S sn S% eSe n S@e n 2SD n K ' f
@ n SS H%2 n eSD ' f DD G
# H 2S. n f@ n 2K2 e@2 n SSH n eK2
e@2 n SSH n eK2 SS. H n 2SH
$
f
i||ih : The constraint given by
;A AA AA AA AA
? AA AA AA AA A=
@ %2c K %22c De G
;A
? A=
e@ S n @D n S2 eK n S sn S% eSe n S@e n 2SD n K ' f
@ n SS H%2 n eSD ' f DD G
# H 2S. n f@ n 2K2 e@2 n SSH n eK2
e@2 n SSH n eK2 SS. H n 2SH
$
f
is convex (better, it is an LMI). Its projector can be built using convex techniques.
VW +a,AW 6 wWt
A matrix of V? is RJr|e re6_es?|e (PSD), denoted by f, if
;3 5 U?c 3T3 f
AiLhi4: The set of PSD matrices is convex.
hLLu:
Vn? '
5 V?m;3 5 U?c 3T3 f ' _
35Uq
5 V? m 3T3 f
' _
35Uq
;?
= 5 V? m [
c
55@ f
<@
>
For instance, the constraint F E%c %2 given by 3
EC
n % n .%2 e n D% %2 S n 2% n %2
e n D% %2 2 n D% n D%2 b n D% n %2
S n 2% n %2 b n D% n %2 e n % n H%2
4
FD f
is convex, because it is the reciprocal image of the convex set Vn by the @ss?e function:
# %
%2
$
$ 3 EC
n % n .%2 e n D% %2 S n 2% n %2
e n D% %2 2 n D% n D%2 b n D% n %2
S n 2% n %2 b n D% n %2 e n % n H%2
4 FD
The projection of an LMI with 6 variables requires 26 LMI optimizations.
Each of them can be performed by SeDuMi which has a worst-case complexity of
?D6 n ?2D62
*L}
0
