Steigungen hoherer Ordnung zur verifizierten globalen by Marco Schnurr

By Marco Schnurr

In dieser Arbeit wird die automatische Berechnung von Steigungstupeln zweiter und h?herer Ordnung behandelt. Die entwickelten Techniken werden in einem Algorithmus zur verifizierten globalen Optimierung verwendet. Anhand von Testbeispielen auf einem Rechner wird der neue Algorithmus mit einem Algorithmus aus der Literatur verglichen.

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Extra info for Steigungen hoherer Ordnung zur verifizierten globalen Optimierung German

Example text

X0 )j , xj+1 , . . , xn       xj − (x0 )j δf (x; x0 )ij := ❢ür xj = (x0 )j        cij ❢ür xj = (x0 )j ❢ür ❜❡❧✐❡❜✐❣❡s cij ∈ R ❡✐♥❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 ✳ ❋❛❧❧s f (x0 ) ❡①✐st✐❡rt✱ s♦ s❡t③❡♥ ✇✐r cij := lim xj →(x0 )j δf (x; x0 )ij = ∂fi (x0 )1 , . . , (x0 )j , xj+1 , . . , xn . ∂xj ■♥❞❡♠ ✇✐r ❢ür δf ([x] ; x0 )ij ❥❡✇❡✐❧s ❡✐♥ ■♥t❡r✈❛❧❧ ✇ä❤❧❡♥✱ ❞❛s ❞✐❡ ▼❡♥❣❡ δf (x; x0 )ij | x ∈ [x] ❡✐♥s❝❤❧✐❡ÿt✱ ❡r❤❛❧t❡♥ ✇✐r ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ δf ([x] ; x0 ) ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ✷✹ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ❊✐❣❡♥s❝❤❛❢t❡♥ ❋ür ❞✐❡ ❇❡r❡❝❤♥✉♥❣ ❡✐♥❡r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ δf ([x] ; x0 ) ❡rst❡r ❖r❞♥✉♥❣ ❤❛❜❡♥ ✇✐r ③✉✈♦r ✈❡rs❝❤✐❡❞❡♥❡ ▼ö❣❧✐❝❤❦❡✐t❡♥ ❦❡♥♥❡♥❣❡❧❡r♥t✱ ♥ä♠❧✐❝❤ ❞✐❡ ❆✉s✇❡rt✉♥❣ ❞❡r ❏❛❝♦❜✐✲ ▼❛tr✐①✱ ❞✐❡ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ✈♦♥ 1 f (x0 + t (x − x0 )) dt 0 ✉♥❞ ❞✐❡ ▼❡t❤♦❞❡ ♥❛❝❤ ❍❛♥s❡♥✳ ❆♥❤❛♥❞ ❞❡s ❢♦❧❣❡♥❞❡♥ ❇❡✐s♣✐❡❧s ✇♦❧❧❡♥ ✇✐r ❞✐❡ ❧❡t③✲ t❡r❡♥ ❜❡✐❞❡♥ ▼❡t❤♦❞❡♥ ✈❡r❣❧❡✐❝❤❡♥✳ ❇❡✐s♣✐❡❧ ✶✳✸✳✷✼ ❲✐r ❜❡tr❛❝❤t❡♥ ❞✐❡ ❋✉♥❦t✐♦♥ f : [x] ⊆ R2 → R2 , f = (f1 , f2 ) , ♠✐t f1 (x1 , x2 ) = x21 + 1 x22 , f2 (x1 , x2 ) = x1 .

0, 1, 0, . . , 0)t ∈ Rn ✱ i = 1, . . n✳ ❉✐❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❡rst❡r ❖r❞♥✉♥❣ δf [x]i,− ; x0 − si ei ∈ IRn×n ✈♦♥ f ❛✉❢ [x]i,− ❙❛t③ ✶✳✸✳✸✹ Rn ✱ ❜❡③ü❣❧✐❝❤ x0 − si ei ✉♥❞ δf [x]i,+ ; x0 + ti ei ∈ IRn×n ✈♦♥ f ❛✉❢ [x]i,+ ❜❡③ü❣❧✐❝❤ x0 + ti ei s❡✐❡♥ ❣❡❣❡❜❡♥✱ ✉♥❞ A ∈ Rn×n s❡✐ ❡✐♥❡ ♥✐❝❤ts✐♥❣✉❧är❡ ▼❛tr✐①✳ ❲✐r s❡t③❡♥ [l]i,+ := A f x0 + ti ei i A δf [x]i,+ ; x0 + ti ei + j=i ij · [−sj , tj ] ✉♥❞ [l]i,− := A f x0 − si ei i A δf [x]i,− ; x0 − si ei + j=i ij · [−sj , tj ] . ●✐❧t ❢ür ❥❡❞❡s i ∈ {1, . .

X2 − 0 ❉❛♠✐t ❡r❣✐❜t s✐❝❤ x1 x22 1 δf (x; x0 ) = x2 0 ✉♥❞ δf ([x] ; x0 ) = ❇❡♠❡r❦✉♥❣ ✶✳✸✳✷✽ ③✳ ❇✳ ③✉ [−1, 1] 1 [−1, 1] 0 . ➘♥❞❡rt ♠❛♥ ✐♥ ❍❛♥s❡♥s ▼❡t❤♦❞❡ ❞✐❡ ❡✐♥❣❡❢ü❣t❡♥ ❙✉♠♠❛♥❞❡♥ fi x1 , . . , xn − fi (x0 )1 , . . , (x0 )n = fi x1 , . . , xn − fi x1 , . . , xn−1 , (x0 )n + fi x1 , . . , xn−1 , (x0 )n −fi x1 , . . , xn−2 , (x0 )n−1 , (x0 )n + fi x1 , . . , xn−2 , (x0 )n−1 , (x0 )n − + · · · + fi x1 , (x0 )2 , . . , (x0 )n − fi (x0 )1 , . . , (x0 )n xj − (x0 )j . ❉✉r❝❤ P❡r♠✉t❛t✐♦♥ ❡r❣❡❜❡♥ s✐❝❤ ❡♥ts♣r❡❝❤❡♥❞ ❢ür ❡✐♥❡ ❋✉♥❦t✐♦♥ f : Rn → Rn ✐♥ ❥❡❞❡r ❑♦♠♣♦♥❡♥t❡ fi ❣❡♥❛✉ n!

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