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Kliknij, aby edytować styl wzorca tytułu A global optimization method for solving parametric linear system whose input data are rational functions of interval parameters Iwona Skalna AGH University of Science and Technology Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Iwona Skalna, Poland Small Workshop on Interval Methods’09, Lausanne Revised affine arithmetic Evolutionary optimization Examples Conclusions

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A global optimization method for solving parametric linear system whose input data are

rational functions of interval parameters

Iwona Skalna

AGH University of Science and Technology

Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Iwona Skalna, Poland Small Workshop on Interval Methods’09, Lausanne

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

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Parametric linear systems

Optimization problem

Interval global optmization

Monotonicity test

Revised affine arithmetic

Evolutionary optimization

Examples

Conclusions

Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

)(),()( pqpp bxA Parametric linear system

is defined as a family of real linear systems

qp p,ppppp ),(),()( bxA

),()(

),()(

0 qfqb

pfpa

ii

ijij

where )(ijf are nonlinear continuous

functions of parameters

with coefficients

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

The goal is to find the thightest interval enclosure for S, possibly the interval hull solution defined as

SSYSYS sup,inf| nIR

Parametric (united) solution set is define as

)()(|),( qbxpAqpRxS n qpqpS

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

If the solution is monotone with respect to all parameters, then the interval hull solution can be calculated by solving at most 2n real linear systems with coefficients being the respective endpoints of interval parameters

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

In general case, to calculate the hull solution, the following 2n constrained optimization problems must be solved

),,(min qpxi

qpqp

where ii qbpAqpx )()(),( 1is an objective function

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

),,(max qpxi

qpqp

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

ni ,,1

The optimizations problems are solved using an interval global optimization. The interval global optimization algorithm has the following steps:

Various acceleration techniques are used to speed up the convergence of global optimization. The monotonicity test is the most important one for the considered problem.

Step 1. Initialize the list L =(pq, x(pq))

Step 2. Remove (pq, x(pq)) from the list L

Step 3. Bisect pq = pq1pq2

Step 4. Calculate x(pqi), pqi

Step 5. Perform the monotonicity test

Step 6. If w(pq) < STOP else GOTO 2

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

),,()(),(

)( qppqp

p xp

A

p

xA

ii

If a devirative has constant sing, then the corresponding interval can be reduced to one of its edges.

ii q

b

q

xA

)(),()(

qqpp

0)(

),(

i

x qpor 0

)(

),(

i

x qp

The monotonicicty test is performed using the Direct Method solving parametric linear systems. To check the sign of derivatives, the following parametric linear systems must be solved:

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmizationMonotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

The Direct Method is also used to calculate inclusion function for the objective function x(p,q) is

Direct Method requires affine-linear dependencies. The nonlinear functions must be transformed into affine-linear forms. This is acheived using the revised affine arithmetic.

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Arithmetic operations used in this work are defined as follows:

Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

Revised affine form

]1,1[,,)()(ˆˆ1

00

rirrr

n

iiii yxyxyxyx

]1,1[,,ˆ1

0

rirr

n

iii xxxx

]1,1[,,||)(ˆ1

0

rirr

n

iii xxxx

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

multiplication

],1,1[,,||5.0

)(5.0ˆˆ

1

100

100

rir

n

iiirrrr

n

iiii

n

iii

yxuvuyvxyx

xyyxyxyxyx

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

where

n

ii

n

ii yvxu

11

||,||

reciprocal

division

),0(,2

2

2

21ˆ1

10

yyy

yyyyy

yy

y

yy

yyyyy

yyy rri

n

i

i

,ˆ1

ˆ

ˆ

0

0

yp

y

x

y

x

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

),0(,2

2

2

21ˆ1

10

yyy

yyyyy

yy

y

yy

yyyyy

yyy rri

n

i

i

where ],[ yyy is a range of an affine form y

where rrri

n

iii yxxy

y

xxp || 0

1 0

0

Interval global optimization method produces hull solution forparametric linear systems with affine-linear dependencieswhich is en enclosure for the solution set of the original systemwith non-affine dependencies. The amount of the overestimation is verified using an evolutionary optimization method. Eachevolutionary algorith has the following steps:

Step 1. Initialize population P(t : 0)

Step 2. Crossover P(t)

Step 3. Mutation P(t)

Step 4. Select P(t+1) from P(t)

Step 5. t : t + 1

Step 6. If done then STOP else GOTO 2

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

The results of evolutionary optimization depends strongly onparameters. Here, the following parameters:

Population size: 16Number of generations: 80Crossover probability: 0.1Mutation probability: 0.9

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

and the following genetic operators are used :

non-uniform mutationarithmetic crossover

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

Example 1. Two dimensional systems with 5 parameters

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

]5.0,48.0[,],96.1,92.1[],98.0,96.0[, 54231 ppppp

Example 1. Two dimensional systems with 5 parameters

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

]5.0,48.0[,],96.1,92.1[],98.0,96.0[, 54231 ppppp

Example 3. Real-life problem of structure mechanics

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

One-bay structural steel frame

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland

Global optimization method can be succesfully used for solving parametric linear systems whose input data are rational functions of interval parameters

The main drawback of global optimization is the complexity. This deficiency can be overcome by parallel programming techniques

The parallelism can be introduced both in the process of the monotonicity check and in the optimization process. This will be the subject of future work

OutlineParametric linear systems

Optimization problemGlobal optmization

Monotonicity test

Revised affine arithmeticEvolutionary optimizationExamplesConclusions

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Small Workshop on Interval Methods’09, LausanneIwona Skalna, Krakow, Poland