3 rd nsf workshop on imprecise probability in engineering analysis & design february 20-22, 2008...

3RD NSF Workshop on Imprecise Probability in Engineering Analysis & DesignFebruary 20-22, 2008 | Georgia Institute of Technology, Savannah, USA

On using global optimization for approximating hull solution of parametric interval systems

Iwona Skalna

AGH University of Science & Technology, Poland

Andrzej Pownuk

The University of Texas, El Paso, USA

2


Outline

Parametric interval linear systems

Global optimization Monotonicity test

Subdivision directions

Multidivision

Examples

Concluding remarks

accelerationtechniques

3


Parametric linear systems

)(),()( qbqpxpA

,)(1

0

k

ijijij ppa

kk

iii IRppqb

p,)(1

000

)(),(:,|),( qbxqpAqpxqpSS qp

Parametric solution set

ySIRySSS n |sup,inf

Interval hull solution

Parametric linear system with parameters p, q

Coefficients: affine linear functions of parameters

4


Optimization problem (hull solution)

niqpxi

qp

,,1),,(min

qp

Let A(p) be regular, pIRk, and ximin, xi

max denote the global

solutions of the i-th minimization

and, respectively, maximization problem

Then the interval vector

Theorem

niqpxi

qp

,,1),,(max

qp

),(],[],[ 1maxminmaxmin qpSxxxxxn

iii

5


Global optimization – general assumptions

Define rIRk+l with ri = pi for i = 1, , k, and rj = qj-k, j = 1, , k.

Then x(p,q) can be written in a compact form as x(r).

w(f(x)) 0 as w(x) 0

Inclusion function f = xi(p,q) is calculated using a Direct Method

for solving parametric linear systems. It can be easily shown that

the Direct Method preserves the isotonicity property, i.e. x y

implies f(x) f(y). It is also assumed that for all inclusion functions holds:

6


Global optimization - algorithm

Step 0. Set y = r and f = min x(y). Initialize list L = {f, y} and cutoff

level z = max x(y)Step 1. Choose a coordinate v {1, , k+l} for subdivision

using one of the subdivision rules Step 2. Bisect (multisect) y in direction v: y1 y2 , int(y1)int(y2) =

Step 3. Calculate x(y1) and x(y2) and set fi = min x(yi) and

z = min { z, max x(y1) , max x(y1) }Step 4. Remove (f, y) from the list LStep 5. Cutoff test: discard any pair (fi, yi) if fi > zStep 6. Monotonicity testStep 7. Add any remaining pairs to the list, if the list become empty

then STOPStep 8. Denote the the pair with the smallest first element by (f*, y*) Step 9. If w(y*) < then STOP, else goto 1.

,,)int()int(,1

s

ijii jiyyy

7


Monotonicity test

The monotonicity test is used to figure out whether the function f is

strictly monotone in a whole subbox y x. Then, y cannot contain a

global minimizer in its interior. Therefore, if f satisfies

Monotonicity test is pefromed using the Method for Checking the Monotonicity (MCM). The MCM method is based on a Direct Method for solving parametric linear systems. Let f = x(p, q). Approximations of partial derivates are obtained by solving the following k+l parametric linear systems

0)(0)(

yyii x

f

x

f

,,,1),()( * kmxbp

xpA m

m

where x* is an approximation of the solution set S calculated using Direct method, and

lrxbq

xpA r

r

,,1),()( *

.*** ,,,1,*,)( x xnjbxxb jrr

jjmjmm

j

8


Subdivision direction selectionThe following merit function (Ratz, Csendes) is used to define subdivision rules:

where D(i) is determined by a given rule.

)(max)(and},,1{|min: iDjDnjjki

Rule A (Hansen):

Rule B (Casado):

Rule C (Walster, Hansen):

D(i) = w(yi)

D(i) = w(F(yi)w(yi))

D(i) = p(fk, fi)

Lyffk

ff

ffffp

lll

kk

),(|min:

),(

9


Example 1

Young’s modulus Y = 7.01010[Pa]Cross section aresa C = 0.003[m2]Length L = 2[m]Loads P1 = P2 = P3 = 30[kN]The stiffness of all bars is uncertain by 5%

10


Results – nodes displacementsn x lb[10-5] x ub[10-

5]n x lb[10-5] x lb[10-

5u

1 -32.53 -29.27 12 3.90 4.67

2 -1.61 -1,45 13 -16.23 -14.67

3 -26.45 -23.93 14 3.18 3.87

4 -2.41 -2.17 15 -3.63 -2.96

5 -15.78 -14.27 16 3.18 3.87

6 -1.69 -1.37 17 -0.05 0.05

7 -4.08 -3.37 18 2.35 3.02

8 -0.96 -0.57 19 -0.46 -0.40

9 0.36 0.50 20 0.85 1.47

10 3.90 4.67 21 -2.78 -2.09

11 -26.45 -23.93result of the Global Optimization = result of the Evolutionary Algorithm

11


Example 2

result of the Global Optimization = result of the Evolutionary Algorithm

Young’s modulus Y = 2.11011[Pa]Cross section aresa C = 0.004[m2]Length L = 2[m]Loads P1 = 80[kN], P2 = 120[kN]The stiffness of thick bars is uncertain by 5%

12


Conclusions

Global Optimization Method can be succesfully used to approximate hull solution of parametric linear systems

Monotonicity test significantly improves the convergence of Global Optimization

Multidivision and different rules for subdivision directions have no influence on the convergence of the Global Optimization, they are computationally very expensive

In future work we will try combine different methods for solving parametric linear systems

3 rd nsf workshop on imprecise probability in engineering analysis & design february 20-22, 2008...

Documents