Transcript To STOP or not to STOP

To STOP or not to STOP
A question in Global Optimization
By I. E. Lagaris
August, 2005
Department of Computer Science, University of Ioannina
Ioannina - GREECE
1
Contributions
Research performed in collaboration with
Ioannis G. Tsoulos .

PhD candidate, Dept. of CS, Univ. of Ioannina
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2
Searching for “Local Minima”
f (x)
One-Dimensional Example
Exhaustive procedure:
From left to right
minimization-maximization
repetition.
S
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x
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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
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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: Ai  {x : x  S , LS ( x)  yi }
 The RA depends strongly on the local search
(LS) procedure.
 The measure of an RA is denoted by
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m(Ai).
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ASSUMPTIONS …

Deterministic local search.
Implies non-overlapping basins.
Ai  Aj  , S   Ai , m(S )   m( Ai )
i
i

Sampling is based on the uniform distribution.
Implies that a sampled point belongs to Ai with
m( Ai )
probability:
i 

m( S )
There is no zero-measure basin, i.e.
m( Ai )  0
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Coverage based stopping rule
If m( Ai ) can be calculated, then a rule may be
formulated based on the space coverage:

c
w
i 1
m( Ai )
m( S )
w,
being the number of
minima discovered so far.
i.e.: STOP when c→1
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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:
m( Ai ) Li

m( S ) L
w
note that:
L  L
i 1
i
Unfortunately this estimation is useless in the present
framework, since it will always yield: c=1
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Double Box
Consider a box S2 that contains S and satisfies:
m(S2 )  2  m(S )
Sample points from S2, and perform local
searches only from points contained in S.

c
w
i 1
m( Ai )
m( S )

2
w
i 1
m( Ai )
m( S 2 )

2
w
i 1
Li
L
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).
kN
ck  2
M k and
1 k
 c  k   ci → 1
k i 1
 k2 (c)  c2 k   c 2k → 0
2
2
|

c


1
|


(
c
)

(
c
)

p

and
 STOP if:
k
k
k
last (c)
p  (0,1)
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last indicates the iteration
during which the latest
minimum was discovered
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p
0.3
0.5
Function
Minima
Calls
Minima
Shubert
400
1150243
400
Gkls(3,30)
30
961269
Rastrigin
49
Test2N(5)
0.7
Calls
0.9
Minima
Calls
Minima
577738
400
322447
395
139768
29
302583
23
41026
15
3920
50384
49
19593
49
13581
49
10034
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
Multistart performance with Double Box, for a range of
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Calls
p values
12
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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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)
i
L
( w1)
i
L
m( Ai )
 (nw  1)
m( S )
An estimation that may be used is:
m( Ai )

m( S )
Li
w
L
k 1
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k

Li
w
n
k 1
k
14
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:
L ( K )  L ( K  1) 
( w)
i
( w)
i
Li
w
K  nj
, L(i w) (0)  L(i w)
j 1
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The Observables’ criterion
The quantity:
 ( w)
1  L j (K )  L j
E ( w, K )   
w
w j 1 
 Ll
l 1

w





2
Tends asymptotically to zero.
Hence, STOP if:
2
E(w, K )   k (E), and  k2 (E)  p last
( E)
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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
If
 k stands for the probability to recover yk
then the probability of finding
l 1
l minima in m trials is given by:
P  (1    i ) P
l
m
i 1
Probability that a new
minimum is found other
than y1 , y2 ,, yl 1
August, 2005
in a single trial,
l 1
m 1
l
 (  i ) P
i 1
l
m 1
Note that:
0
1
P 0
P 1
1
1
Probability that one of the
first l minima y1 , y2,, yl
is found again.
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Expected values
We use the estimate:
The expected value for the
number of minima,
estimated after m trials is
given by:
m
 w  m   lP
l 1
l
m
m( Ak ) L(km)
k 

m( S )
m
The corresponding variance
is given by:
m
2
2 l
2
m
m
m
l 1
 (w)   l P   w 
The RULE
STOP if: | w m wm |  m (w)
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2
and  m2 (w)  p last
(w)
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Other rules
Uncovered fraction of space:
STOP if:
w( w  1)
L( L  1)
w( L  1)
west 
Lw2
Probability all minima are found:
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Boender &
Rinnooy Kan (1987)
west  w  12
L 1  i
PAll  
i 1 L  1  i
w
STOP if:
Zieliński (1981)
c(w)  
Estimated number of
minima:
STOP if:
c( w) 
Boender &
Romeijn (1995)
PAll  1   ,   0

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MULTISTART
Uncovered
fraction
FUNCTION
CAMEL
RASTRIGIN
SHUBERT
HANSEN
GRIEWANK2
GKLS(3,30)
GKLS(3,100)
GKLS(4,100)
GUILIN(10,200)
GUILIN(20,100)
Test2N(4)
Test2N(5)
Test2n(6)
Test2n(7)
GOLDSTEIN
BRANIN
HARTMAN3
HARTMAN6
SHEKEL5
SHEKEL7
SHEKEL10
August, 2005
MIN
6
49
400
527
528
16
34
20
191
96
16
32
64
128
4
3
3
2
5
7
10
Estimated # of
minima
FC
5642
38104
316640
426056
565932
5286
11464
6010
354650
263869
17373
37639
81893
175850
5906
2173
3348
3919
8720
11742
16020
MIN
6
49
400
527
529
13
61
12
200
100
16
32
64
128
4
3
3
2
5
6
10
FC
2549
121182
8034563
14220225
18941546
4249
97124
7816
4736609
1760826
18716
78931
336353
1435579
3812
1782
2750
3851
4733
5485
10611
Double Box
Observables
Expected # of
minima
MIN
MIN
MIN
6
49
400
527
529
29
97
95
200
100
16
32
64
128
4
3
3
2
5
7
10
FC
5503
19593
577738
612015
1765175
302853
7492103
8629052
3351391
1906288
19424
30607
34840
117953
5391
1856
3509
3903
22128
30702
36838
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6
49
400
527
528
23
94
73
200
100
16
32
64
128
4
3
3
2
5
7
9
FC
2720
13342
369958
391597
996188
84291
5658721
5290564
2178890
973307
5296
10700
27679
70370
3842
1782
2778
3907
6430
7581
9812
6
49
400
527
527
25
92
93
199
99
16
32
64
128
4
3
3
2
5
7
10
FC
2916
9007
212353
240092
449090
96260
3416276
6358587
1136783
655374
3970
7707
18367
41981
3850
1782
2772
3851
8850
10914
12751
21
TMLSL
Uncovered
fraction
FUNCTION
CAMEL
RASTRIGIN
SHUBERT
HANSEN
GRIEWANK2
GKLS(3,30)
GKLS(3,100)
GKLS(4,100)
GUILIN(10,200)
GUILIN(20,100)
Test2N(4)
Test2N(5)
Test2N(6)
Test2N(7)
GOLDSTEIN
BRANIN
HARTMAN3
HARTMAN6
SHEKEL5
SHEKEL7
SHEKEL10
August, 2005
MIN
4
31
52
66
295
8
10
2
139
74
12
32
20
104
4
3
3
2
4
4
5
Estimated # of
minima
FC
406
1821
2971
3837
56348
908
1171
310
76928
72794
874
3798
1513
10776
583
354
1567
685
446
436
412
MIN
4
15
236
361
48
7
7
2
181
99
8
32
11
128
3
3
3
2
4
4
5
FC
404
1614
226048
451662
298174
624
727
310
1600657
734645
508
7750
1184
73947
587
359
1567
677
446
434
406
Double Box
MIN
6
49
400
527
528
28
91
78
200
100
16
32
64
128
4
3
3
2
5
6
9
FC
3415
24123
358623
811679
1929165
191867
6683608
12269342
3470206
1915478
3821
8953
54555
88750
4606
1007
2014
912
6434
14617
18447
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Observables
MIN
6
49
400
527
527
28
90
72
200
100
16
32
64
128
4
3
3
2
5
7
10
FC
7325
12815
321031
781209
1602467
364849
2535356
11199668
2650971
1333144
2807
6634
34649
69371
11709
182919
18271
87124
90356
161969
78645
Expected # of
minima
MIN
6
49
400
527
526
29
98
93
199
99
16
32
64
128
4
3
3
2
5
6
9
FC
1815
9199
275773
501844
591899
483227
32034155
70398347
1752886
867312
4162
6092
27500
52248
2522
1227
3857
2094
3310
4363
10288
22
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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