??sa???? st? ?e?t?st?p???s?

1
??sa???? st? ?e?t?st?p???s?
2
???sµ??
?e?t?st?p???s? ? d?ad??as?a e?a??st?p???s?? ?
µe??st?p???s?? t?? ep???µ?t?? s????t?s?? ??st???,
e?? ta?t?????a ??a??p?????ta? ?f?st?µe???
pe?????sµ?? (Belegundu). ???t?st?? s?ed?asµ??
? s?ed?asµ?? µ?a? ?atas?e???, ? ?p???? ??a??p??e?
t?? ?atas?e?ast???? p??d?a??af?? ?a? t??
?e?t???????? apa?t?se??, e?? ta?t?????a
e?a??st?p??e? s???e???µ??a ???t???a, ?p?? e??a?
s?????ste?a t? ??st?? ?a? t? ß???? t?? ?atas?e???.

• ? ße?t?st?p???s? ß???? t??? µ??a?????? ?a
• s?ed??s??? ap?d?t???te?a s?st?µata
• s?ed??s??? ????te?? a???ß? s?st?µata ?a?
• a?apt????? ?a???t?µe? µe??d??? ??a t? ße?t??s?
  t?? ep?d?s?? ?pa????t?? s?st?µ?t??

• ??p???? efa?µ???? ße?t?st?p???s??
• s?ed?asµ?? ae??s?af?? e????st?? ß?????
• ße?t?st?p???s? d???se?? ?a? ????ß?? se ???µata
• ß??t?st?? s?ed?asµ?? ??e?t????? ?????µ?t??
• ße?t?st?p???s? s?ed?asµ?? pa?a?????

3
?????t???? ???sµ?? ?
?e????? s?ed?as?? n-d??stat?? ?a?tes?a??? ?????
s??teta?µ???? ?p?? ???e ????a? a?t?p??s?pe?e? µ?a
µetaß??t? s?ed?as?? xi (i1,..n). S?µe??
s?ed?as?? ??a s?µe?? st? ???? s?ed?as?? p??
a?t?p??s?pe?e? µ?a d??at? ? ad??at?
ep???s? ?etaß??t?? s?ed?as?? ?e? µp????? ?a
ep??e????? t??a?a, p??pe? ?a p?????? s???e???µ??a
?e?t??????? ?a?a?t???st??? ?a? ?a pa??????
ap?de?t? s?ed?as? ?e?????sµ?? s?ed?as?? ??
pe?????sµ?? p?? p??pe? ?a p????? µ?a s?ed?as?
?ste ?a e??a? ap?de?t?. ?p?????? d?? ?at?????e?
pe?????sµ??

• ?e?t???????? a?t?p??s?pe???? pe?????sµ???
  s?µpe??f???? ?a? ep?d?s?? t?? s?st?µat??
• Ge?µet????? f?s???? pe?????sµ?? t?? µetaß??t??
  s?ed?as??

4
?????t???? ???sµ?? ??
??t??e?µe???? s????t?s? ? s????t?s? ??st??? ??a
???t???? s?????s?? t?? ap?de?t?? ??se?? µe ß?s?
t? ?p??? ???eta? ? ep????? t?? ß??t?st??
s?ed?as??. ?ta? ?p?????? pe??ss?te?e? ap? µ?a
?t??e?µe????? s??a?t?se?? t? p??ß??µa ???eta?
multi-objective. ?p?f??e?a pe?????sµ??
?p?f??e?a, st?? pe????? s?ed?as??, p?? ??????? ??
t?µ?? t?? µetaß??t?? s?ed?as?? ? x ?a?
??a??p????? t? s?????? gi(x) 0 ?? s????? t??
ep?fa?e??? pe?????sµ?? gi(x) 0, i1,.m p??
d?a?????e? t?? ap?de?t? pe????? e??a? ???st? ??
s???et? ep?f??e?a pe?????sµ??. ?p?f??e?a
s????t?s?? ??st??? ??t?st???? t?? ep?f??e?a?
pe?????sµ??.
?ta? s?ed?ast??? a?t?? ?? ep?f??e?e? st?? pe?????
s?ed?as?? µp??e? ?a e?t?p?ste? ??af??? t?
ß??t?st? s?µe??
5
?e?????sµ?? ?a? s?µe?a s?ed?as??
S?µe?a s?ed?as?? p?? ß??s???ta? st?? ep?f??e?a
pe?????sµ?? gi(x) 0 ??a??p????? ???a?? t???
pe?????sµ??? ? ep?f??e?a a?t? ?????e? t?? pe?????
s?ed?as?? se d?? ?????? gi(x) lt 0 ef??t? ?a?
ap?de?t? gi(x) gt 0 µ?-ap?de?t?
s?µe?a s?ed?as??
??a s?µe?? s?ed?as?? p?? ß??s?eta? se µ?a ?
pe??ss?te?e? t?t??e? ep?f??e?e? ?a?e?ta? e?e????
pe?????sµ? ??a s?µe?? s?ed?as?? p?? de?
ß??s?eta? se ep?f??e?e? pe?????sµ?? ?a?e?ta?
e?e??e?? s?µe??
6
??at?p?s? p??ß??µat?? ße?t?st?p???s??
? p?e????f?a t?? µ??a????????? p??ß??µ?t??
af????? st?? e?a??st?p???s? µe?e??? µe
pe?????sµ???
??t?, st? ?e???? µ??f? µ?-??aµµ????
p????aµµat?sµ??, e?f?????ta? ??
??a??st?p???s? t?? f(x) p?? ?p??e?ta? st???
pe?????sµ??? gi(x) 0 i 1,.......,m ?a?
hj(x) 0 j 1,......., l
?
??a??st?p???s? t?? f(x), x ? O, ?p?? O x g
0 , h 0 ?a? O ? ef??t? pe?????

• ?p??
• x(x1, x2,......., xn)t d????sµa st???
  n-p?a?µat???? µetaß??t?? s?ed?as??
• f a?t??e?µe???? s????t?s? ? s????t?s? ??st???
• g pe?????sµ?? a??s?t?ta? ?a?
• h pe?????sµ?? ?s?t?ta?

7
?a?at???se??

• ?? a??s?t???? pe?????sµ?? s?µpe???aµß????? ?µµesa
  ta a??tata ?a? ?at?tata ???a t?? µetaß??t??
  s?ed?as??
• ??? t? p??ß??µa de? ??e? pe?????sµ??? ? ef??t?
  pe????? e??a? ???? ? ????? x ? Rn.
• ? a?t??e?µe???? s????t?s? ?a? ?? pe?????sµ?? st?
  ??aµµ??? p????aµµat?sµ? e??a? ??aµµ????
  s??a?t?se?? t?? x
• ? a?t??e?µe???? s????t?s? sta p??ß??µata
  tet?a??????? p????aµµat?sµ?? e??a? tet?a??????
  s????t?s? t?? x, e?? ?? pe?????sµ?? e??a?
  ??aµµ???? s??a?t?se?? t?? x
• ?e????? f???? ?? pe?????sµ?? e?f?????ta? se
  ?a??????p???µ??? µ??f? ?ste ?a e?asfa??ste? ?t? ?
  t?µ? t??? ß??s?eta? st? d??st?µa 0,1. p.?. e??
  ? t?s? p?? a?apt?sseta? se µ?a d?µ? p??pe? ?a
  e??a? µ????te?? ap? 10000 psi t?te

8
?at?????e? ße?t?st?p???s??
? Lucien Schmit t? 1960 ?ta? ? p??t?? p??
a?a?????se t? d??at?t?ta s??d?asµ?? te??????
ße?t?st?p???s?? µe t? d?µ??? s?ed?as?
9
?e?t?st?p???s? µe??????

• ? s????t?s? ??st??? f, s??????, e??a? t? ß????
  t?? ?atas?e???
• ?? pe?????sµ?? gi af????? sta ???a t?s?? ?a?
  µetat?p?s??
• ?? µetaß??t?? s?ed?as?? µp????? ?a e??a?

???t??µa eµßad?? d?at?µ?? ??ßd?? ?p?ped?
e?tat??? ?at?stas? ? d?µ? ?e??f??? p???? ???e
?.S. ??at?µ? d???? ??p? ad???e?a?
?e?t?st?p???s? s??µat??

• ?a p??ß??µata a?t? as??????ta? µe t?? ?a????sµ?
  t?? s?????? e??? s?µat??, t? s??µa t??, t?
  µ??e??? ?p?? ?.?.
• ?? s??teta?µ??e? t?? ??µß?? t?? ?.S µetaß?????ta?
  epa?a??pt???
• ? ßas??? ?d?a e??a? ? pa?aµet??p???s? t??
  p???µat?? (s?s?et?sµ?? t?? s??teta?µ???? t??
  ??µß?? t?? p???µat?? µe t?? pa?aµ?t???? t?? ?.S.)

?e?t?st?p???s? t?p?????a?
?a p??ß??µata a?t? as??????ta? µe t?? ?ata??µ?
t?? ??????, t? d?µ??????a ?p?? ?a? t? s??µat?sµ?
? t?? e???e??? st???e??? t?? d?µ??
10
????d?? ?e?t?st?p???s??

• ?? ßas???te?e? µ???d?? ße?t?st?p???s?? ???????ta?
  se t?e?? ?at?????e?
• ?a??µat???? te?????? (?tete?µ???st???? µ???d??)
  ???s?µ?p?????ta? ??a t?? e?t?p?sµ? t?? e????st??
  s??a?t?se?? a??et?? µetaß??t?? µe pe?????sµ???.
  ?? µa??µat???? µ???d?? p???????ta? ap? t??
  ep?st?µ?????? pe?????? t?? ??????µ????
  µa??µat???? ?a? t?? ep??e???s?a??? ??e??a? ?a?
  ?ta? ?? p??te? p?? efa?µ?st??a? se p??ß??µata
  ß??t?st?? s?ed?asµ?? ?atas?e???. p.?.
  µ?-??aµµ???? p????aµµat?sµ??, tet?a???????
  p????aµµat?sµ??, SQP, ?.?.
• St??ast???? µ???d?? ???s?µ?p?????ta? ??a t??
  a????s? p??ß??µ?t?? t??a??? µetaß??t?? µe ???st?
  ?ata??µ?. ?at?????a a?t?? ap?te???? ??
  ??e???t???? ? ?a?ß??e?e? µe??d??? p?? e??a?
  te?????? ßas?sµe?e? se t??a?e? d?ad??as?e?
  de??µat?????a? p?? p??s?µ??????? ?e?et????
  d?e??as?e? ?a? p??t?pa ??a t?? ep???s?
  p??ß??µ?t?? ße?t?st?p???s?? p.? ?e?et????
  a??????µ??, µ???d?? simplex, p??sµ????µe??
  a??pt?s?
• Stat?st???? µ???d?? ???s?µ?p?????ta? ??a t??
  d?µ??????a eµpe?????? µ??t???? ap? pe??aµat???
  st???e?a ?ste ?a ep?te???e? a???ß?ste??
  a?apa??stas? t?? f?s???? p??ß??µat??. ?.?.
  Regression analysis

11
???t?st? s?ed?as?
??ad??as?a p??s????s?? ß??t?st?? p?????t??
? ß??t?st? s?ed?as? ep?t?????eta? µe µe??d??? p??
pa?????ta? ap? ta s??????a ????sµ??? pa??ta ?a?
ta ?p??a ßas????ta? st? µ???d? t?? ?epe?asµ????
St???e???.

• ?e???? ep?t???????ta?
• ????ß?ste??? ?p?????sµ??
• ?µes? a???????s?
• ?e?t??s? t?? p?????t??

12
???t? St?d?? ?e?t?st?p???s??

• ??at?p????ta? a?????? ?d?e? ??a t?? p??a??
  d??ta?? t?? ?atas?e???
• ?a???????ta?
• ? ?e?t?????a
• t? ??st?? t?? d?ad??as?a? pa?a?????
• t? ??st?? t?? p?????s?? t?? p?????t??
• ??et??eta? ?a? e?t?µ?ta? µe???? p????? µ??t????
• S????????ta?
• d??f??a ?????
• µ???d?? pa?a?????
• ?e?µet????? d?at??e??
• ?ata???f??ta? pa?ad???? p?? µp??e? ?a af????? t??
  e??te????? s?????e?, t?? e?de??µe?? ep?p??s?et?
  f??t?s? ?.?.

?e???? ???eta? µe??t? ?a? ?ata??af? µe?e??? p??
e?d?af????? t? ???st? ??a ???e p??a?? pe??pt?s?
t?? ?atas?e???. (p.?. ß????, µ???st? t?µ? t??
t?s?? von Mises)
13
Local Sensitivity Study
??e???? p?? ?ata???e? st? ?at? p?s? ? a??a?? t??
t?µ?? µ?a? pa?aµ?t??? (p.?. t? ????? t??
µ??t????, ?? d?ast?se?? t?? ?.t.?.) e??a?
s?µa?t??? ??a t?? ?atas?e??
??ad??as?a

• ?aµß??eta? ? t?µ? µ?a? pa?aµ?t??? (p.?. t? µ????
  t?? µ??t????)
• ??te?e?ta? ??e??a ??a t? s?µpe??f??? ??p????
  ?a?a?t???st???? µe?e??? ??a t?µ?? 0.5 t??
  ??f?e?sa? t?µ?? (p.?. t? ß???? ??µ???, ? µ???st?
  a?apt?ss?µe?? t?s? von Mises ?.?.)
• ?atas?e???eta? ??a d????aµµa a??µesa st??
  pa??µet?? ?a? ??p??? ?a?a?t???st??? µ??e??? ?p??
  µ?a e??e?a ??aµµ? e???e? ta 2 s?µe?a p?? ?????
  p?????e?. ?? ? ???s? t?? e??e?a? e??a? µe?a??te??
  ap? 1, t?te ? pa??µet??? a?t? ep??e??e? s?µa?t???
  t? s?µpe??f??? t?? µ??t???? (d??. ?t? µ?a µ????
  a??a?? se µ?a pa??µet?? ( 0.5 ) a????e? a?s??t?
  ??p??? µ??e???).

Global Sensitivity Study
??t? t? st?d?? pa???e? ????????µ??? ?a? e??????
e????a t?? ?atas?e??? a??????ta? µ?a pa??µet??,
ef?s?? ? ??e??a ???eta? ??a ??? t? e???? t??
t?µ?? p?? µp??e? ?a p??e? ? pa??µet???.
?p?t??esµa ???f?µa ?p?? eµfa???eta? ?e???a?a ?
e??st?te t?µ? ??a ??p??? µ??e???, ?a??? a????e? ?
t?µ? t?? pa?aµ?t???.
14
?e?te?? St?d?? ?e?t?st?p???s??

• Optimization Study (ße?t?st?p???s?) ??????ta?
• ? a????? ?e?µet??a t?? µ??t????
• ? st????
• ?? pe?????sµ??
• ?? pa??µet??? s?ed?as??

?a?at???se??

• ?a ap?te??sµata t?? Local Sensitivity Study
  µp??e? ?a e??a? pa?ap?a??t???, af?? ? a??a?? µ?a?
  pa?aµ?t??? de? e??a? p??ta ??aµµ??? se s??s? µe
  ??p??? µ??e???.
• ?? Global Sensitivity study ap?te?e? t?? ?a??te??
  ?a? asfa??ste?? t??p? ?a d?asf???s?? ?t? ??
  pe?????sµ?? e??a? ??????. S??ep??, ?a
  e?asfa??ste? ?t? µ?sa sta ???a t?? pe?????sµ??
  ?p???e? t??????st?? µ?a ??s?.
• ?? pa??µet??? s?ed?as??, p?? ?a???????ta?, e??a?
  a?t?? p?? ????? e?t?p?ste? st? 1? St?d?? ?a?
  ep?d???? s?µa?t??? st? s?µpe??f??? t??
  ?atas?e???.
• ? pe?????sµ?? t?? a???µ?? t?? pa?aµ?t???
  s?ed?as?? ??e? ?? ap?t??esµa t? µe??s? t?? ??????
  p?? apa?te?ta? p???e?µ???? ?a ?????????e? ?
  ??e??a t?? ße?t?st?p???s??.

15
????d?? ?e?t?st?p???s??

• ?tete?µ???st??? Sequential Quadratic Programming
  - SQP
• ??e??a? se p??t?p? Simplex Method
• ??e??a? se p??t?p? Hooke Jeeves
• St??ast??? Genetic Algorithms
• St??ast??? Simulated Annealing

• ??e??e?t?µata st??ast???? a??????µ??
  ße?t?st?p???s?? ??a?t? ?tete?µ???st????
• ??ep???ast?? ap? a?????? s?????e? ??a µe?a??te?e?
  p??a??t?te? e??es?? t?? ap???t?? ß??t?st?? - de?
  pa??de???ta? se t?p??? ß??t?sta
• ?e? ??t??? pe?????sµ??? ??a t?? ???? ??se??
• ???a? e?sta?e??
• ?e???e?t?µata st??ast???? a??????µ??
  ße?t?st?p???s??
• ????? µ???? ta??t?ta s?????s??

16
St??ast???? ????d??

• ???s?µ?p????? t??a???? a???µ??? ??a ?a ?a??????
  t?? ???? a?a??t?s??.
• ?e? e??a? ap?te?esµat???? µ???d?? ?a? de?
  ßas????ta? se ??p??? µa??µat??? µ??t???.

?p???ste?? ?d?a st??ast???? µe??d??
?st? ?(?1,?2,,??) ?a? ?i t??a??? st? d??st?µa
-1,1 ??? f(xa? )lt f(x), t? s?µe?? xa?
???eta? ap?de?t?. ?p????eta? ??? ? ?a?
epa?a?aµß??eta? ? d?ad??as?a. ??? µet? ap?
a??et?? epa?a???e?? de? ?p???e? p???d??, ? t?µ?
t?? a µe???eta? ?a? ? d?ad??as?a ?a?a????e?.
17
Sequential Quadratic Programming SQP

• ????d?? 2?? t???? ??a p??ß??µata µe pe?????sµ???
• 3 ????a ß?µata
• ??a???s? t?? ?ss?a??? µ?t???? t?? s??a?t?se??
  Lagrange
• ?p???s? t?? p??ß??µat?? tet?a???????
  p????aµµat?sµ??
• ??e??a ??aµµ?? ?a? ?p?????sµ?? t?? s??a?t?se??
  ???d???
• ?e ded?µ??? t? ?e???? p??ß??µa, ? ?e?t???? ?d?a
  e??a? ? d?aµ??f?s? e??? tet?a???????
  ?p?p??ß??µat?? ßas?sµ??? st?? tet?a??????
  p??s????s? t?? ?a???a???a??? s??a?t?se??
• ?, ?a???a???a??? s??te?est??
• g, pe?????sµ?? t?? p??ß??µat??

18
Hook Jeeves
????d?? µ?de????? t??e?? ????? pa?a??????

• ???? a?a??t?s?? ap? s?µe?? B (s?µe?? ß?s??)
• ??e?e???s? ???? ap? t? s?µe?? B µe ß?µa s (0.8).
• ??? ? e?e?e???s? de? e??a? ep?t???? s s0.5
• ??? ? e?e?e???s? e??a? ep?t???? ????eta?
  d?e????s? ??e??a? BP (xp-xB)
• P e??a? t? ??? s?µe?? ß?s?? B. ?? a????? s?µe?? B
  t??eta? B.
• ? BB' epe?te??eta? st? d?e????s? a?a??t?s?? st?
  s?µe?? E ?? (xE2xB-xB').
• ??e?e???s? ???? ap? t? s?µe?? E.
• ??? ? e?e?e???s? de? e??a? ep?t????, ss0.5
• ??? ? e?e?e???s? e??a? ep?t???? ????eta?
  d?e????s? ??e??a? BE (xE-xB)
• ??? t? ß?µa s lt 1E-08 te?µat?sµ?? t?? a??????µ??,
  a????? ep?st??f? st? ß?µa 2.

19
? µ???d?? SIMPLEX (????t?p??)

• ?? ????t?p? ? SIMPLEX st? ???? RN, e??a? ??a
  ?atas?e?asµa ?1 s?µe???.
• ?? ????f?? t?? p???t?p?? a?t?st?????? se
  d?a??sµata t?? RN ?a? s?µß??????ta? ??
  W0,W1,,WN.
• St? ???? t?? 2 d?ast?se?? e??a? t??????, st? ????
  t?? 3 d?ast?se?? e??a? tet??ed??, ???.

??ad??as?e? ?a? p???e?? SIMPLEX

• ?e??t??d?t?s? ????µ?s? t?? ????f?? Wi ?ts? ?ste
  fi lt f(Wi) lt f(Wj) lt fj ??a iltj
• ??t?st??f? ?? s?µe?? r e??a? t? a?t?st??f? t?? w
  ?? p??? c r c(c - w).
• ???tas? ?? s?µe?? e p????pte? ap? t?? d??tas?
  t?? r ?? p??? c e c 2(r - c)..
• S???????s? ?? s?µe?? k p????pte? ap? t??
  s???????s? t?? r ?? p??? c
• k c (r-c)/2.
• S?st??? ?? p???t?p? s?st??eta? p???? t?? W0 Wi
  (W0 Wi )/2 ??a i1,2,,N

20
?e????af? a??????µ?? SIMPLEX
21
???s?µ????µe?? ???pt?s? - Simulated Annealing (SA)
???pt?s? ??ad??as?a ?at? t?? ?p??a ??a µeta?????
a?t??e?µe?? ?e?µa??eta? se µ?a ?e?µ???as?a ?a?
st? s????e?a ???eta? stad?a?? ???s?µ????µe??
a??pt?s? p??ß??µata ße?t?st?p???s?? ep?????ta?
a????????ta? t?? ?d?a d?ad??as?a t?? µet?ßas??
ap? µ?a ?at?stas? ?s????p?a? se µ?a ????,
ft????ta? ?ts? st?? e????st? e????e?a t??
s?st?µat??.
?e????af? a??????µ?? S? ?

• ???sµ?? µetaß??t?? a?????? T (1E-01) , e????st??
  Tmin, ? a???µ?? ?p?????sµ?? s????t?s? ??st??? ??a
  sta?e?? T (70), µ???st?? a???µ?? ep?t???µ????
  d???µ?? a?? T (30) ?a? µ???st?? a???µ??
  ap?t???µ???? d???µ?? (1000).
• ?p?????sµ?? t?? s????t?s?? ??st??? fmin se a?????
  s?µe??
• ???sµ?? t?? µet??t? KK1 (counter)

22
?e????af? a??????µ?? S? ??

• G??es? ????, t??a??? s?µe??? ?a? ?p?????sµ?? t??
  s????t?s?? ??st??? fs.
• ??? fs lt fmin fmin fs
• ??af??et??? efa?µ????ta? Metropolis criteria
  , ?? fs fmin
• ??? ? ??a ep????? ap????f?e? ??a ??? s?µe??
  ep????eta?.
• ??e???? s?????s?? ?a? e?te st? 3 ß?µa e?te st?
  ß?µa 7
• ??? K lt Kmax ep?st??f? st? ß?µa 3, a????? st?
  ß?µa 8.
• T0.8T
• ??e???? s?????s?? ?a? e?te st? 3 e?te t????

23
???? ?e?t?????a? ?e?et???? a??????µ??
?????aµµa ???? t?p???? ?e?et???? a??????µ??

• ??d???p???s? t?? ????? ??se?? se ????d?a.
• ?atas?e?? ???µ?s?µ?t?? ap? ta ????d?a.
• ?atas?e?? e??? a?????? p????sµ?? t??a???
  ???µ?s?µ?t??.
• ?p?????sµ?? a?t??e?µe????? s????t?s?? ??a ??a ta
  µ??? t?? p????sµ??.
• ?p?????, ep???asµ?? ?a? µet???a?? ??? ?t??
  d?µ?????????? a??et? ??a ???µ?s?µata ?ste ?a
  ap?te???? µ?a ??a ?e???.
• ??t??at?stas? pa???? p????sµ?? ap? t?? ??a ?e???
  ?a? epa?????? a??????µ??.