Metódy vol nej optimalizácie

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1 Matematické programovanie Metódy vol nej optimalizácie p. 1/35

2 Informácie o predmete Informácie o predmete p. 2/35

3 Informácie o predmete METÓDY VOL NEJ OPTIMALIZÁCIE Prednášajúca: M. Trnovská (M 267) Cvičiaci: J. Komadel Informácie týkajúce sa predmetu na stránke Konzultácie sa dajú dohodnút mailom. trnovska@pc2.iam.fmph.uniba.sk p. 3/35

4 Informácie o predmete Podmienky hodnotenia Domáce úlohy na cvičeniach 40 % Výstup: matlabovský kód, spracovanie úlohy (výstupu, komentárr, grafy, tabul ky) do pdf súboru (latex) Skúška 60 % Minimálna kostra - základné znalosti Písomka Ústna čast (nepovinná) p. 4/35

5 Informácie o predmete Podmienky hodnotenia % známka A B C D E p. 5/35

6 Informácie o predmete Literatúra Základná literatúra: M. Hamala, M. Trnovská: Nelineárne programovanie (1. čast ) Doplnková literatúra: S. Boyd, L. Vandenberghe: Convex optimization p. 6/35

7 Informácie o predmete Harmonogram prednášok Úvod do predmetu Motivačné prendášky Metódy minimalizácie funkcie jednej premennej Metódy intervalovej aproximácie Metódy bodovej aproximácie Metódy minimalizácie funkcie n premenných Klasické metódy Gradientné metódy Metóda CSR Newtonova metóda Moderné metódy Združené gradienty Kvázinewtonovské metódy Metódy pre vel korozmerné úlohy p. 7/35

8 Matematické programovanie Matematické programovanie p. 8/35

9 Matematické programovanie ÚLOHA MATEMATICKÉHO PROGRAMOVANIA min f 0 (x) x K R n (MP) Ak K = R n alebo K je otvorená množina - hovoríme o úlohe na vol ný extrém. Ak K je uzavretá množina spravidla popísaná systémom rovníc a nerovníc K = {x R n f i (x) 0,i I, h j (x) = 0,j J}, hovoríme o úlohe s ohraničeniami. p. 9/35

10 Matematické programovanie Z hl ladiska typu funkcií rozlišujeme Úlohu lineárneho programovania - ak funkcie f 0,f i, i I, h j, j J sú lineárne (afínne); Úlohu nelineárneho programovania - ak aspoň jedna z funkcií f 0,f i (x), i I, h j (x), j J nie je lineárna; Úlohu konvexného programovania - ak funkcie f 0,f i (x) 0,i I sú konvexné a funkcie h j (x), j J sú lineárne (afínne); LINEÁRNE KONVEXNÉ NELINEÁRNE p. 10/35

11 Matematické programovanie Klasifikácia úloh nelineárneho programovania: Ak K = {x R n h j (x) = 0,j J}, - klasická Lagrangeova úloha na viazaný extrém Ak K = {x R n f i (x) 0,i I}, - úloha nelineárneho programovania programovania v užšom zmysle Ak K = {x R n f i (x) 0,i I, h j (x) = 0,j J}, - úloha nelineárneho programovania programovania v širšom zmysle p. 11/35

12 Matematické programovanie METÓDY VOL NEJ OPTIMALIZÁCIE = metódy riešenia úloh na vol ný extrém min f(x) x R n (U1) kde f : R n R a často sa predpokladá diferencovatel nost (1. alebo 2. rádu). Optimálne riešenie úlohy (U1) - ˆx R n Optimálna hodnota f(ˆx) = ˆf Platí x R n : f(ˆx) = ˆf f(x) p. 12/35

13 Matematické programovanie Ak f je diferencovatel ná, ˆx je optimálne riešenie, tak zrejme f(ˆx) = 0. (NP) Ak f je navyše konvexná, podmienka (NP) je nie len nutnou, ale aj postačujúcou podmienkou optimality - riešenie úlohy (U1) je ekvivalentné riešeniu systḿu (NP). Úloha (U1) sa zvyčajne rieši nejakým iteračným algoritmom, ktorý generuje postupnost bodov x 0,x 1,x 2,... R n, f(x k ) ˆx, pre k Algoritmus končí v ε-presnom riešení, t. j. ked f(x k ) ˆf < ε. alebo ked je splnené iné kritérium suboptimality - napr. f(x k ) < ε. p. 13/35

14 Matematické programovanie Niektoré úlohy možno priamo naformulovat ako úlohu na vol ný extrém (U1) - napr. min f(x) = Ax b x R n - hl adanie približného riešenia systému Ax b v norme. min f(x) = Ax b +λ x x R n - hl adanie približného riešenia systému Ax b s regularizáciou. p. 14/35

15 Matematické programovanie Riešenie mnohých úloh matematického programovania s ohraničeniami min f 0 (x) f i (x) 0, i I h j (x) = 0, j J (MP) možno transformovat na riešenie postupnosti úloh na vol ný extrém (U1): Metódy Lagrangeovych funkcií Metódy vnútorného bodu p. 15/35

16 História optimalizačných úloh p. 16/35

17 Euklidova úloha: Do daného trojuholníka ABC vpíšte rovnobežník ADEF tak, že AF DE, AD EF a jeho obsah je maximálny. Ako naformulujeme Euklidov problém ako optimalizačnú úlohu? p. 17/35

18 Hx(c x) max f(x) = x (0,c) c maximum sa nadobúda pre x = c 2. Euklid to ukázal geometrickou úvahou dnes vieme, že stačí riešit f (x) = 0. Fermat ( ) Newton ( ) Leibnitz ( ) Lagrange ( ) p. 18/35

19 Joseph Louis Lagrange ( ) "Essai sur d une nouvelle méthode pour déterminer les maxima et minima des formules intégrales indéfinies" p. 19/35

20 Lagrangeova úloha: Min{f 0 (x) h i (x) = 0, i = 1,...,m} kde f 0,h j : R n R, j = 1,...,m Lagrangeova funkcia: L : R n R m R, L(x,u) = f 0 (x)+ m u i h i (x) i=1 Lagrangeova metóda: x L(x,u) = 0, u L(x,u) = 0. - systém nelineárnych rovníc - Newtonov algoritmus p. 20/35

21 Moderná matematická teória nelineárneho porgramovania Konferenica: Second Berkeley Symposium on Mathematical Statistics and Probability, 31. júl august, 1950 Miesto: Statistical Laboratory of the University of California, Berkeley Albert W. Tucker (Princeton) prezetnoval príspevok s názvom Nonlinear Programming (spoluautor: Harold W. Kuhn) Prvý krát sformulovaná úloha typu min g(x) f i (x) 0, i = 1,...,m x 0 a sformulovaná a dokázaná Kuhn-Tuckerova veta Na úvodnej strane: This work was done under contracts with the Office of Naval Research. p. 21/35

22 Harold W. Kuhn a Albert W. Tucker p. 22/35

23 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) p. 23/35

24 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne p. 24/35

25 lineárne vs. nelineárne 3 x- 2 y 4 x 2-2 x x 6 + x y- 4 y y y x y x p. 25/35

26 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne p. 26/35

27 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne George Dantzig - simplexová metóda p. 27/35

28 George Dantzig ( ) - Simplexová metóda p. 28/35

29 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne George Dantzig - simplexová metóda Narendra Karmarkar - metódy vnútorného bodu p. 29/35

30 Narendra Karmarkar ( ) - Metódy vnútorného bodu p. 30/35

31 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne George Dantzig - simplexová metóda Narendra Karmarkar - metódy vnútorného bodu konvexné vs. nekonvexné p. 31/35

32 konvexné vs. nekonvexné 0.5 x y 2 4 x 2-2 x x 6 + x y- 4 y y y x y x p. 32/35

33 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne George Dantzig - simplexová metóda Narendra Karmarkar - metódy vnútorného bodu konvexné vs. nekonvexné p. 33/35

34 Posúvanie hranice medzi jednoduchými a zložitými úlohami z historického hl adiska Matematické programovanie - teória a algoritmy na riešenie optimalizačných úloh typu min f 0 (x) x K R n (MP) lineárne vs. nelineárne George Dantzig - simplexová metóda Narendra Karmarkar - metódy vnútorného bodu konvexné vs. nekonvexné Nesterov & Nemirovski - metódy vnútorného bodu pre konvexné úlohy. p. 34/35

35 Yurii Nesterov a Arkadi Nemirovski p. 35/35

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