Telecommunicatie en Informatieverwerking UNIVERSITEIT GENT Didactisch materiaal bij de cursus Academiejaar Tel: 09/ Fax: 09/ Prof. dr. ir. W. Philips Optimalisatietechnieken

© W. Philips, Universiteit Gent, versie: 5/3/ b. 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 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 . 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. Philips Department of Telecommunications and Information ProcessingFax: University of GentTel: St.-Pietersnieuwstraat 41, B9000 Gent, Belgium

Duale lineaire programma’s

© W. Philips, Universiteit Gent, versie: 5/3/ b. 5 Dual-based phase-1 Nieuwe methode om een initieel extreem punt te zoeken: door het aanpassen van de winstfunctie van het primaal probleem forceert men een geldige basis in het aangepast duaal probleem door oplossen van aangepast duaal probleem verkrijgt men het optimum x ’ van het aangepast primaal probleem het resultaat x ’ is niet optimaal voor het origineel probleem, maar wel een extreem punt

© W. Philips, Universiteit Gent, versie: 5/3/ b. 6 Voorbeeld … Noch in het primaal, noch in het duaal probleem is de huidige basisoplossing een extreem punt Duaal: Primaal: Standaardvorm P1P yyyz yyyz yyy    max Standaardvorm D1D1

© W. Philips, Universiteit Gent, versie: 5/3/ b. 7 …Voorbeeld … Door de primale winstfunctie te veranderen kan men van de basisoplossing in het duaal probleem steeds een extreem punt maken Het gebied van mogelijke oplossingen van het primaal probleem verandert niet maar dat van het duaal probleem verandert wel Duaal: Primaal: Standaardvorm P1P yyyz yyyz yyy    max Standaardvorm D1D1 +1

© W. Philips, Universiteit Gent, versie: 5/3/ b. 8 …Voorbeeld … Oplossen aangepast probleem met duale simplexmethode geeft: duaal optimum: y 1 = z 1 = y 2 =0 y 3 =1 z 2 =4 primaal optimum: w 3 = x 2 =0 w 1 =18  0 x 1 =7  0 w 2 =6  0 Ter herinnering: het aangepast primaal probleem heeft een andere winstfunctie dan het opgegeven probleem maar het zelfde gebied van mogelijke oplossingen Dus: het primaal optimum van het aangepast probleem kan dienen als initieel extreem punt voor het niet-aangepast primaal probleem w 3 = x 2 =0 w 1 =18  0 x 1 =7  0 w 2 =6  0 P1P yyyz yyyz yyy    max +1 D1D1

© W. Philips, Universiteit Gent, versie: 5/3/ b. 9 Lagrangiaanse dualiteit Komt aan bod in een latere les