To STOP or not to STOP

August, 2005 Department of Computer Science, University of Ioannina Ioannina - GREECE

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To STOP or not to STOP

By I. E. Lagaris

A question in Global Optimization


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Contributions

Research performed in collaboration with Ioannis G. Tsoulos .

PhD candidate, Dept. of CS, Univ. of Ioannina


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Searching for “Local Minima”

One-Dimensional Example

Exhaustive procedure:

From left to right minimization-maximization repetition.

)(xf

xS


4

Searching for “Local Minima”

Two-Dimensional Example

“Egg holder”

The exhaustive technique used in one-dimension, is not applicable in two or more dimensions.

S Level plots in 2-D


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The “MULTISTART ” algorithm

Sample a point x from S Start a local search, leading to a minimum y If y is a new minimum, add it to the list of minima Decide “ to STOP or not to STOP ” Repeat

If the decision is right, the iterations will not stop before all minima inside the bounded domain S are found.


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The “Region of Attraction” (RA)

The set of all points that when a local search is started from, concludes to the same minimum.

Formally:

The RA depends strongly on the local search (LS) procedure.

The measure of an RA is denoted by m(Ai).

})(,:{ ii yxLSSxxA


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ASSUMPTIONS … Deterministic local search. Implies

non-overlapping basins.

Sampling is based on the uniform distribution.

Implies that a sampled point belongs to Ai with probability:

There is no zero-measure basin, i.e.

i

ii

iji AmSmASAA )()(,,

)(

)(

Sm

Am ii

0)( iAm


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Coverage based stopping rule

i.e.: STOP when c→1

)(

)(1

Sm

Amc

w

i i w, being the number of minima discovered so far.

)( iAmIf can be calculated, then a rule may be formulated based on the space coverage:


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Estimating m(Ai)

Let L be the number of the performed local searches

and Li those that ended at yi.

An estimation then, may be obtained by:

Unfortunately this estimation is useless in the present

framework, since it will always yield: c=1

L

L

Sm

Am ii )(

)(

w

ii LL

1

note that:


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Double BoxConsider a box S2 that contains S and satisfies:

)(2)( 2 SmSm Sample points from S2, and perform local

searches only from points contained in S.

L

L

Sm

Am

Sm

Amc

w

i i

w

i i

w

i i 1

2

11 2)(

)(2

)(

)(

L, now stands for the total number of sampled points.


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Implementation Keep sampling from S2 until N points in S are

collected. (N =1 for Multistart) At iteration k, let Mk be the total number of

sampled points (kN of them in S).

kk M

kNc 2 and

k

iik c

kc

1

1→ 1

222 )( kkk ccc → 0

last indicates the iteration during which the latest minimum was discovered

STOP if: )()( 22 cpc lastk )(|1| cc kk and)1,0(p


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Function Minima Calls Minima Calls Minima Calls Minima Calls

Shubert 400 1150243 400 577738 400 322447 395 139768

Gkls(3,30) 30 961269 29 302583 23 41026 15 3920

Rastrigin 49 50384 49 19593 49 13581 49 10034

Test2N(5) 32 78090 32 30607 32 20870 32 13462

Test2N(6) 64 85380 64 34840 64 22535 64 15393

Guilin(20) 100 3405112 100 1906288 100 854511 71 79331

Shekel-10 10 93666 10 36838 10 23780 10 15976

p 0.3 0.5 0.7 0.9

Multistart performance with Double Box, for a range of p values


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Observables rule This rule relies on the agreement of values of

observable (i.e. measurable) quantities, to their expected asymptotic values.

The number of times Li that minimum yi is found, is compared to its expected value.

yi are indexed in order of their appearance. Hence y1 requires one application of the LS, y2 requires additional n2 applications, y3 additional n3 …

Let the number of the recovered minima so far be denoted by w.


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)(

)()1()1()(

Sm

AmnLL i

ww

iw

i

Expectation values

The expectation value of the number of times the ith minimum is found, at the time when the wth minimum is recovered for the first time, is recursively given by:

w

kk

iw

kk

ii

n

L

L

L

Sm

Am

11

)(

)(An estimation that may be used is:


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Keep trying …

Suppose that after having found w minima, there is a number (say K) of consecutive trials without any success, i.e. without finding a new minimum.

The expected number of times the ith minimum is found at that moment is given recursively by:

)()(

1

)()( )0(,)1()( wi

wiw

jj

iwi

wi LL

nK

LKLKL


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The Observables’ criterion

The quantity:

2

1

1

)()(

1),(

w

jw

l lL

jLKwjL

wKwE

Tends asymptotically to zero.

Hence, STOP if:

)()(),(),( 22 EpEandEKwE lastkk


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“Expected Minimizers” Rule

Based on estimating the number of local minima inside the domain of interest.

The estimation is improving as the algorithm proceeds.

The key quantity is the probability that l minima are found after m trials.

Calculated recursively.


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Probabilities

lm

l

ii

lm

l

ii

lm PPP 1

1

11

1

1

)()1(

kIf stands for the probability to recover in a single trial,kythen the probability of finding l minima in m trials is given by:

Probability that a new minimum is found other than 121 ,,, lyyy

Probability that one of the first l minimais found again.

lyyy ,,21,

1

01

1

01

P

PNote that:


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Expected valuesThe expected value for the number of minima,

estimated after m trials is given by:

m

l

lmm lPw

1

2

1

22 )( m

m

l

lmm wPlw

The corresponding variance is given by:

m

L

Sm

Am mkk

k

)(

)(

)(

We use the estimate:

STOP if: )()()(|| 22 wpwandwww lastmmmm The RULE


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Other rules

)1(

)1()(

LL

wwwcUncovered fraction of space: Zieliński (1981)

STOP if: )(wc

Estimated number of minima: 2

)1(

wL

Lwwest

Boender & Rinnooy Kan (1987)

STOP if: 21 wwest

Probability all minima are found:

w

iAll iL

iLP

1 1

1 Boender & Romeijn (1995)

STOP if: 0,1 AllP


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Uncovered fraction

Estimated # of minima

Double Box Observables Expected # of minima

MULTISTART

FUNCTION MIN FC MIN FC MIN FC MIN FC MIN FCCAMEL 6 5642 6 2549 6 5503 6 2720 6 2916RASTRIGIN 49 38104 49 121182 49 19593 49 13342 49 9007SHUBERT 400 316640 400 8034563 400 577738 400 369958 400 212353HANSEN 527 426056 527 14220225 527 612015 527 391597 527 240092GRIEWANK2 528 565932 529 18941546 529 1765175 528 996188 527 449090GKLS(3,30) 16 5286 13 4249 29 302853 23 84291 25 96260GKLS(3,100) 34 11464 61 97124 97 7492103 94 5658721 92 3416276GKLS(4,100) 20 6010 12 7816 95 8629052 73 5290564 93 6358587GUILIN(10,200) 191 354650 200 4736609 200 3351391 200 2178890 199 1136783GUILIN(20,100) 96 263869 100 1760826 100 1906288 100 973307 99 655374Test2N(4) 16 17373 16 18716 16 19424 16 5296 16 3970Test2N(5) 32 37639 32 78931 32 30607 32 10700 32 7707Test2n(6) 64 81893 64 336353 64 34840 64 27679 64 18367Test2n(7) 128 175850 128 1435579 128 117953 128 70370 128 41981GOLDSTEIN 4 5906 4 3812 4 5391 4 3842 4 3850BRANIN 3 2173 3 1782 3 1856 3 1782 3 1782HARTMAN3 3 3348 3 2750 3 3509 3 2778 3 2772HARTMAN6 2 3919 2 3851 2 3903 2 3907 2 3851SHEKEL5 5 8720 5 4733 5 22128 5 6430 5 8850SHEKEL7 7 11742 6 5485 7 30702 7 7581 7 10914SHEKEL10 10 16020 10 10611 10 36838 9 9812 10 12751


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FUNCTION MIN FC MIN FC MIN FC MIN FC MIN FCCAMEL 4 406 4 404 6 3415 6 7325 6 1815RASTRIGIN 31 1821 15 1614 49 24123 49 12815 49 9199SHUBERT 52 2971 236 226048 400 358623 400 321031 400 275773HANSEN 66 3837 361 451662 527 811679 527 781209 527 501844GRIEWANK2 295 56348 48 298174 528 1929165 527 1602467 526 591899GKLS(3,30) 8 908 7 624 28 191867 28 364849 29 483227GKLS(3,100) 10 1171 7 727 91 6683608 90 2535356 98 32034155GKLS(4,100) 2 310 2 310 78 12269342 72 11199668 93 70398347GUILIN(10,200) 139 76928 181 1600657 200 3470206 200 2650971 199 1752886GUILIN(20,100) 74 72794 99 734645 100 1915478 100 1333144 99 867312Test2N(4) 12 874 8 508 16 3821 16 2807 16 4162Test2N(5) 32 3798 32 7750 32 8953 32 6634 32 6092Test2N(6) 20 1513 11 1184 64 54555 64 34649 64 27500Test2N(7) 104 10776 128 73947 128 88750 128 69371 128 52248GOLDSTEIN 4 583 3 587 4 4606 4 11709 4 2522BRANIN 3 354 3 359 3 1007 3 182919 3 1227HARTMAN3 3 1567 3 1567 3 2014 3 18271 3 3857HARTMAN6 2 685 2 677 2 912 2 87124 2 2094SHEKEL5 4 446 4 446 5 6434 5 90356 5 3310SHEKEL7 4 436 4 434 6 14617 7 161969 6 4363SHEKEL10 5 412 5 406 9 18447 10 78645 9 10288

Uncovered fraction

Estimated # of minima

Double Box Observables Expected # of minima

TMLSL


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Conclusions

The new rules improve the performance at least for problems in our benchmark suite.

Proper choice of the parameter p, for different methods is important.

Remains to be seen if performance is also boosted in other practical applications.

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To STOP or not to STOP