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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