Telecommunicatie en Informatieverwerking UNIVERSITEIT GENT Didactisch materiaal bij de cursus Academiejaar 2011-2012

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1 Telecommunicatie en Informatieverwerking UNIVERSITEIT GENT Didactisch materiaal bij de cursus Academiejaar 2011-2012 philips@telin.UGent.be http://telin.UGent.be/~philips/optimalisatie/ Tel: 09/264.33.85 Fax: 09/264.42.95 Prof. dr. ir. W. Philips Optimalisatietechnieken

2 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 2 Copyright notice This powerpoint presentation was developed as an educational aid to the renewed course “Optimisation Techniques” (Optimalisatietechnieken), taught at the University of Gent, Belgium as of 1998. This presentation may be used, modified and copied free of charge for non-commercial purposes by individuals and non-for-profit organisations and distributed free of charge by individuals and non-for-profit organisations to individuals and non-for-profit organisations, either in electronic form on a physical storage medium such as a CD-rom, provided that the following conditions are observed: 1.If you use this presentation as a whole or in part either in original or modified form, you should include the copyright notice “© W. Philips, Universiteit Gent, 1998” in a font size of at least 10 point on each slide; 2.You should include this slide (with the copyright conditions) once in each document (by which is meant either a computer file or a reproduction derived from such a file); 3. If you modify the presentation, you should clearly state so in the presentation; 4.You may not charge a fee for presenting or distributing the presentation, except to cover your costs pertaining to distribution. In other words, you or your organisation should not intend to make or make a profit from the activity for which you use or distribute the presentation; 5. You may not distribute the presentations electronically through a network (e.g., an HTTP or FTP server) without express permission by the author. In case the presentation is modified these requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the presentation, and can be reasonably considered independent and separate works in themselves, then these requirements do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the presentation, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. In particular note that condition 4 also applies to the modified work (i.e., you may not charge for it). “Using and distributing the presentation” means using it for any purpose, including but not limited to viewing it, presenting it to an audience in a lecture, distributing it to students or employees for self-teaching purposes,... Use, modification, copying and distribution for commercial purposes or by commercial organisations is not covered by this licence and is not permitted without the author’s consent. A fee may be charged for such use. Disclaimer: Note that no warrantee is offered, neither for the correctness of the contents of this presentation, nor to the safety of its use. Electronic documents such as this one are inherently unsafe because they may become infected by macro viruses. The programs used to view and modify this software are also inherently unsafe and may contain bugs that might corrupt the data or the operating system on your computer. If you use this presentation, I would appreciate being notified of this by email. I would also like to be informed of any errors or omissions that you discover. Finally, if you have developed similar presentations I would be grateful if you allow me to use these in my course lectures. Prof. dr. ir. W. PhilipsE-mail: philips@telin.UGent.be Department of Telecommunications and Information ProcessingFax: 32-9-264.42.95 University of GentTel: 32-9-264.33.85 St.-Pietersnieuwstraat 41, B9000 Gent, Belgium

3 Didactisch materiaal bij de cursus Academiejaar 2010-2011 philips@telin.UGent.be http://telin.UGent.be/~philips/optimalisatie/ Tel: 09/264.33.85 Fax: 09/264.42.95 Prof. dr. ir. W. Philips Optimalisatietechnieken Telecommunicatie en Informatieverwerking UNIVERSITEIT GENT

4 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 4 Copyright notice This powerpoint presentation was developed as an educational aid to the renewed course “Optimisation Techniques” (Optimalisatietechnieken), taught at the University of Gent, Belgium as of 1998. This presentation may be used, modified and copied free of charge for non-commercial purposes by individuals and non-for-profit organisations and distributed free of charge by individuals and non-for-profit organisations to individuals and non-for-profit organisations, either in electronic form on a physical storage medium such as a CD-rom, provided that the following conditions are observed: 1.If you use this presentation as a whole or in part either in original or modified form, you should include the copyright notice “© W. Philips, Universiteit Gent, 1998- 2001” in a font size of at least 10 point on each slide; 2.You should include this slide (with the copyright conditions) once in each document (by which is meant either a computer file or a reproduction derived from such a file); 3. If you modify the presentation, you should clearly state so in the presentation; 4.You may not charge a fee for presenting or distributing the presentation, except to cover your costs pertaining to distribution. In other words, you or your organisation should not intend to make or make a profit from the activity for which you use or distribute the presentation; 5. You may not distribute the presentations electronically through a network (e.g., an HTTP or FTP server) without express permission by the author. In case the presentation is modified these requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the presentation, and can be reasonably considered independent and separate works in themselves, then these requirements do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the presentation, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. In particular note that condition 4 also applies to the modified work (i.e., you may not charge for it). “Using and distributing the presentation” means using it for any purpose, including but not limited to viewing it, presenting it to an audience in a lecture, distributing it to students or employees for self-teaching purposes,... Use, modification, copying and distribution for commercial purposes or by commercial organisations is not covered by this licence and is not permitted without the author’s consent. A fee may be charged for such use. Disclaimer: Note that no warrantee is offered, neither for the correctness of the contents of this presentation, nor to the safety of its use. Electronic documents such as this one are inherently unsafe because they may become infected by macro viruses. The programs used to view and modify this software are also inherently unsafe and may contain bugs that might corrupt the data or the operating system on your computer. If you use this presentation, I would appreciate being notified of this by email. I would also like to be informed of any errors or omissions that you discover. Finally, if you have developed similar presentations I would be grateful if you allow me to use these in my course lectures. Prof. dr. ir. W. PhilipsE-mail: philips@telin.UGent.be Department of Telecommunications and Information ProcessingFax: 32-9-264.42.95 University of GentTel: 32-9-264.33.85 St.-Pietersnieuwstraat 41, B9000 Gent, Belgium

5 Niet-lineaire programma’s Herhaling

6 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 6 De Taylorreeksontwikkeling Gegeven een scalaire vectorfunctie f ( x ) in een n- dimensionale ruimte In vectornotatie: scalair product Taylorreeksbenadering rond x : de gradient ( n x1 kolomvector) de Hessiaan ( n xn matrix)

7 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 7 Optima van niet-lineaire programma’s x is een lokaal optimum van een niet-lineair programma  in een kleine omgeving van x kunnen we geen enkele stap zetten naar een betere mogelijke oplossing x is een minimum als f ( x+  x ) > f ( x ) voor alle 0<||  x||   x is een maximum als f ( x+  x ) < f ( x ) voor alle 0<||  x||   Taylorreeksbenadering: x is maximum   Voor voldoend kleine  x is de eerste term veel groter dan de andere  en voor met c >0 geldt: Stel dat Dus : x is maximum  en voor alle kleine  x : 0 maar dat kan niet in een maximum!

8 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 8 De gradiëntmethode...  f ( x k ) staat loodrecht op het iso-oppervlak f ( x ) =f ( x k ) door x k f ( x,y ) = cte x y x 0x 0 x 2x 2 x 3x 3 x 4x 4 x 1x 1 f ( x k + p k ) wordt optimaal   f ( x k + p k ) p k = 0 Het volgende punt x k+ 1 voldoet dus aan  f ( x k+ 1 ) p k = 0 Interpretatie: in x k+ 1 raakt de rechte x k + p k aan een iso-oppervlak p 0 =  f ( x 0 )   x k+ 1 =  f ( x k+ 1 ) staat loodrecht op  x k  De gradiëntmethode volgt altijd een zigzaglijn optimum

9 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 9 …Produceren of verder verkopen? Duaal probleem minimaliseer y 1 b 1 +  + y i b i +  + y m b m mits y 1, y 2, , y m  0, en a 11 y 1 +  + a i1 y i +  + a m1 y m  c 1 a 12 y 1 +  + a i2 y i +  + a m2 y m  c 2  a 1 n y 1 +  + a in y i +  + a mn y m  c m Primaal probleem maximaliseer c 1 x 1 +  + c j x j +  + c n x n mits x 1, x 2,..., x n  0, en a 11 x 1 +  + a 1 j x j +  + a 1 n x n  b 1 a 21 x 1 +  + a 2 j x j +  + a 2 n x n  b 2... a m 1 x 1 +  + a mj x j +  + a mn x n  b m minimaliseer b t y over y mits y  0 en A t y  c maximaliseer c t x over x mits x  0 en Ax  b

10 © W. Philips, Universiteit Gent, 1998-2012versie: 23/4/2012 09a. 10 indien voor alle j waarvoor x j  0: en voor alle j waarvoor x j  0: Besluit Primaal probleemDuaal probleem = -voorwaardevrije variabele y i  -voorwaardevariabele y i  0  -voorwaardevariabele y i  0 variabele x j  0 variabele x j  0 vrije variabele x j en voor alle j waarvoor x j vrij is: = -voorwaarde  -voorwaarde  -voorwaarde