Nonlinear optimization and applications

Transcription

1 Nonlnear optzaton and applaton CSI 747 / MAH 689 Intrutor: I. Grva Wedneda 7: : p

2 Contraned optzaton proble : R n R n X I E X Inequalt ontrant I { n R :... } Equalt ontrant E { n R :... p}

3 Equalt ontrant....t. n p aranan p arane ultpler... p Optalt ondton

4 Optalt ondton o the Jaoban Newton baed Pral -Dual ethod : ˆ : ˆ loe enouh to oluton approaton I * * * * * * ˆ ˆ α α

5 Contraned optzaton proble : R n R n X I E X Inequalt ontrant I { n R :... } Equalt ontrant E { n R :... p}

6 Inequalt ontrant... n.t. I * I { 3} { 4} Atve et : Pave et : I I * { : * } { : * > } * β > β > β3 3 4 β * β β β 3 he atve et orrepond to the upport vetor!!!

7 Inequalt ontrant... n.t. n.t. I * Choon the atve et a obnatoral proble!!!

8 Optalt ondton Y o the Jaoban Newton ethod Y C Y Y da da C? to tae nto aount the nonneatvt : How Y C aura near the oluton Poble lo o Challene Copleentart!!!

9 Contraned optzaton proble : R n R n X X Inequalt ontrant X { n R :... } Barrer unton B lo

10 Sequental unontraned nzaton tehnque Fao and MCor n lo where > a barrer paraeter oarth repel the trajetor ro the boundar!!! n.t.

11 Sequental unontraned nzaton wth the lo-barrer unton B Output end.5 e.. Update :... / Update : wth aura ethod b Newton' e.. arn Fnd do a whle e.. Selet Selet aura < < > > > γ γ γ ε ε ε... B C C B da where

12 Nonlnear Realn o ontrant Pola Equvalent proble ψ t n.t. - ψ t the aln paraeter aranan or the equvalent proble Φ ψ ψ t lo t - Moded Barrer Pola ψ t e t - Eponental Kort and Bertea

13 Nonlnear realn prnple ˆ ar n Φ R n ˆ Ψ ˆ Ye ed!!! Ψ da ψ Y da ˆ

14 Nonlnear Realn Method Output end... Update : wth aura ethod b Newton' e.. arn Fnd do or n a whle ood hoe a... Selet Selet aura Φ < > > ψ ε ε ε ε Ψ Φ Y ] da [ ] da [ Ψ Ψ Ψ Φ ψ ψ ψ

15 Nonlnear Realn Method pratal atter ln t t.5 ψ t at bt t. 5

16 Contraned optzaton proble : R n R n X X Inequalt ontrant X { n R :... } Barrer unton B lo

17 Sequental unontraned nzaton tehnque Fao and MCor n > > lo where > a barrer paraeter oarth repel the trajetor ro the boundar!!! n.t.

18 Interor pont ethod B... e Ye C da C Y da

19 Interor-pont ethod e Y o the Jaoban Newton ethod Y e C Y Y da da C Copleentart relaaton Y C e C aura near the oluton Poble lo o

20 Optalt ondton Y o the Jaoban Newton ethod Y C Y Y da da C? to tae nto aount the nonneatvt : How Y C aura near the oluton Poble lo o Challene Copleentart!!!

21 Interor-pont ethod e Y o the Jaoban Newton ethod Y e C Y Y da da C Copleentart relaaton Y C e C aura near the oluton Poble lo o

22 Interor pont ethod Sla : w w... w > n.t. w lo w aranan w lo w are arane ultpler w

23 Interor Pont Method IPM e WYe z w w a vetor o the dual varable the Jaoban o Y da W e... da w

24 w WYe e w I W Y Interor Pont Method the proble the aranan o Hean o a p : α ˆ w w w p : α ˆ d : α ˆ < n n d δ α n n < p w w w δ α { } w WYe w v a Mert Funton:

25 Contraned optzaton proble : R n R n X I E X Inequalt ontrant I { n R :... } Equalt ontrant E { n R : j... p} j zz zz

26 Nonlnear Realn auented aranan Method z z Φ ψ ar n ˆ z R n Φ Ye ˆ ˆ Ψ ˆ ˆ z z

27 Nonlnear Realn auented aranan Method j j j z p j z z z z Output end Update : wth aura ethod b Newton' e.. arn Fnd do or or n a whle ood hoe a Selet Selet aura Φ > < > > ψ ε ε ε ε ε z z Ψ Φ z Y ] da [ ] da [ Ψ Ψ Ψ Φ ψ ψ ψ

28 α or all Eaple p < p Eaple [ ] p [ ] p.d. and < p o uarantee deent:

29 Qua-Newton alorth Eaple BFGS

30 Conjuate Gradent Method

31 runated Newton Method

32 Sequental Quadrat Proran n.t..5 A b A b Q q

33 Equalt ontrant....t. n r aranan r arane ultpler... r Optalt ondton

34 Optalt ondton λ o the Jaoban Newton ethod : ˆ : ˆ loe enouh to oluton approaton I * * * * * * ˆ ˆ α α

35 Sequental Quadrat Proran Fndn equvalent olvn the ollown QP:.t..5 n : varate update pral- dual are ued to and the arane ultpler obtaned : ˆ : ˆ

36 SQP: Inequalt ontrant... n.t. aranan arane ultpler... n.t..5 obtaned and the arane ultpler are ued to update pral- dual ˆ : varate ˆ : here are oe ue to be addreed!!! For e.