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diff --git a/src/components/OttimizzazioneLineare/Max.js b/src/components/OttimizzazioneLineare/Max.js index 546a79a..0ef3ea0 100644 --- a/src/components/OttimizzazioneLineare/Max.js +++ b/src/components/OttimizzazioneLineare/Max.js @@ -1,7 +1,7 @@ -import style from "./Unbounded.less"; +import style from "./Styles.less"; export default function (props) { return ( - {props.children ? props.children : "unbounded"} + {props.children ? props.children : "max"} ); } diff --git a/src/components/OttimizzazioneLineare/Min.js b/src/components/OttimizzazioneLineare/Min.js index 546a79a..fb5e6a7 100644 --- a/src/components/OttimizzazioneLineare/Min.js +++ b/src/components/OttimizzazioneLineare/Min.js @@ -1,7 +1,7 @@ -import style from "./Unbounded.less"; +import style from "./Styles.less"; export default function (props) { return ( - {props.children ? props.children : "unbounded"} + {props.children ? props.children : "min"} ); } diff --git a/src/components/OttimizzazioneLineare/Styles.less b/src/components/OttimizzazioneLineare/Styles.less index 683e252..f9e2514 100644 --- a/src/components/OttimizzazioneLineare/Styles.less +++ b/src/components/OttimizzazioneLineare/Styles.less @@ -1,2 +1,17 @@ -@import "../styles/constants.less"; +@import "../../styles/constants.less"; +.unbounded { + color: @blue; +} + +.unfeasible { + color: @red; +} + +.min { + color: @cyan; +} + +.max { + color: @orange; +} diff --git a/src/components/OttimizzazioneLineare/Unbounded.js b/src/components/OttimizzazioneLineare/Unbounded.js index 2aa16c0..eaf7d3f 100644 --- a/src/components/OttimizzazioneLineare/Unbounded.js +++ b/src/components/OttimizzazioneLineare/Unbounded.js @@ -1,7 +1,7 @@ -import style from "Unfeasible.less"; +import style from "./Styles.less"; export default function (props) { return ( - {this.props.children} + {props.children ? props.children : "unbounded"} ); } diff --git a/src/components/OttimizzazioneLineare/Unfeasible.js b/src/components/OttimizzazioneLineare/Unfeasible.js index 0a6799e..3bfe7f4 100644 --- a/src/components/OttimizzazioneLineare/Unfeasible.js +++ b/src/components/OttimizzazioneLineare/Unfeasible.js @@ -1,6 +1,7 @@ +import style from "./Styles.less"; + export default function (props) { return ( -
-
+ {props.children ? props.children : "unfeasible"} ); } diff --git a/src/components/old/todo.css b/src/components/old/todo.css index e5e2f12..9b3a5c4 100644 --- a/src/components/old/todo.css +++ b/src/components/old/todo.css @@ -1,7 +1,7 @@ .todo { - border: 4px yellow solid; - border-radius: 4px; - padding: 2px; + border: 1px yellow solid; + border-radius: 2px; + padding: 1px; background-color: black; color: yellow; } diff --git a/src/index.less b/src/index.less index 3aa098c..a31b141 100644 --- a/src/index.less +++ b/src/index.less @@ -136,3 +136,7 @@ p:first-child { p:last-child { margin-bottom: 0; } + +b { + color: @accent; +} diff --git a/src/routes/calcoloNumerico.js b/src/routes/calcoloNumerico.js index be6e1ae..e745de3 100644 --- a/src/routes/calcoloNumerico.js +++ b/src/routes/calcoloNumerico.js @@ -13,19 +13,12 @@ export default class CalcoloNumerico extends Component { return (

Calcolo Numerico

- +

silvia.bonettini@unimore.it

- -

- Su appuntamento. -

- - -

E' composto da: @@ -34,10 +27,16 @@ export default class CalcoloNumerico extends Component {

  • 2 domande sugli argomenti teorici
  • 1 domanda di implementazione algoritmo in MATLAB (facoltativa per i punteggi migliori del TBL)
    • -

    - Prossimo appello: - -

    +     + +
      +
      2. +
      3. +
      4. +
      5. +
      6. +
      7. +
    diff --git a/src/routes/ottimizzazioneLineare.js b/src/routes/ottimizzazioneLineare.js index 629e78a..722755e 100644 --- a/src/routes/ottimizzazioneLineare.js +++ b/src/routes/ottimizzazioneLineare.js @@ -9,6 +9,10 @@ import Plus from "../components/old/plus"; import Code from "../components/old/code"; import Timer from "../components/old/timer"; import Image from "../components/Image"; +import Unfeasible from "../components/OttimizzazioneLineare/Unfeasible"; +import Unbounded from "../components/OttimizzazioneLineare/Unbounded"; +import Min from "../components/OttimizzazioneLineare/Min"; +import Max from "../components/OttimizzazioneLineare/Max"; const r = String.raw; @@ -116,6 +120,11 @@ export default class OttimizzazioneLineare extends Component { La funzione da minimizzare/massimizzare, tipicamente indicata con una {r`z`} al termine noto.

    + +

    + Le funzioni del sistema che non sono quella obiettivo. +

    +

    Un modo per rappresentare sistemi in forma standard, anche noto come matrice equivalente completa del sistema. @@ -189,23 +198,27 @@ export default class OttimizzazioneLineare extends Component {

    - Un algoritmo per massimizzare efficientemente variabili di sistemi lineari, derivato da Gauss-Jordan. + Un algoritmo per minimizzare/massimizzare efficientemente variabili di sistemi lineari, derivato da Gauss-Jordan.

    - E' spiegato semplicemente qui. + E' spiegato in modo semplice qui, e ci sono dei codici sorgenti di esempio qui.
    1. Trasforma il sistema in forma standard.
      2. -
    2. Finchè ci sono variabili con coefficienti positivi nella funzione obiettivo: +
    3. Trova tante variabili linearmente indipendenti quante siano le righe: esse saranno la base iniziale.
      4. +
    4. Finchè ci sono variabili con coefficienti positivi/negativi nella funzione obiettivo:
        -
      1. Scegli una variabile della funzione obiettivo, chiamandola variabile entrante. Come? Vedi nel prossimo pannello.
        2. -
      2. Trova la variabile di base (detta variabile uscente) con il valore minore per il rapporto {r`\frac{termine\ noto}{coeff.\ variabile\ entrante}`}
        3. +
      3. + Scegli la prima variabile con coefficiente positivo/negativo nella funzione obiettivo: essa è la variabile entrante. + Si potrebbe scegliere qualsiasi variabile, ma scegliendo sempre la prima possibile (Regola di Bland) ci si assicura che l'algoritmo termini. +
        4. +
      4. Trova la variabile di base (detta variabile uscente) tramite il rapporto {r`\frac{termine\ noto}{coeff.\ variabile\ entrante}`}: scegli la variabile con il rapporto minore, assicurandoti che esso sia positivo. Se tutti i rapporti sono negativi, allora il problema è .
      5. Riscrivi tutte le funzioni del sistema in termini della variabile entrante.
      5. -
    5. Il termine noto della funzione obiettivo è il tuo risultato.
      6. +
    6. I termini noti dei vincoli sono le coordinate del risultato, mentre il termine noto della funzione obiettivo è il valore minimizzato/massimizzato.
     @@ -213,13 +226,15 @@ export default class OttimizzazioneLineare extends Component { Se il problema è rappresentato in forma di tableau, allora esso è risolvibile applicando l'algoritmo di Gauss-Jordan, in aggiunta tenendo conto delle regole per la selezione delle variabili entranti e uscenti.

    - -
      -
    • Coefficiente maggiore nella funzione obiettivo.
      • -
    • Incremento maggiore della funzione obiettivo.
      • -
    • A caso.
      • -
    • Regola di Bland: scegli variabili entranti e uscenti con indice minore (ovvero, prendi le prime possibili). È usato nella teoria perchè impedisce i cicli infiniti!
      • -
    +     + + +

    + Una soluzione con almeno una variabile di valore 0, dovuta a uno o più vincoli ridondanti. +

    +

    + Senza Regola di Bland e in presenza di vincoli ridondanti si rischia di trovarsi a fare pivot infiniti. +

    @@ -233,10 +248,109 @@ export default class OttimizzazioneLineare extends Component {

    - Un estensione del Simplex per permettere la risoluzione di problemi con termini noti negativi. + Un estensione del Simplex per permettere la risoluzione di problemi la cui origine non è una soluzione ammissibile.

    - Prevede l'introduzione di un problema ausiliario. + Prevede l'introduzione di un problema ausiliario, le cui variabili sono dette artificiali e sono solitamente rappresentate come {r`y_n`}. +

    + + E' spiegato in modo semplice qui. + +     + +
      +
    1. Crea un nuovo tableau, aggiungendo variabili artificiali in modo da avere una base ammissibile.
      2. +
    2. Sostituisci la vecchia funzione obiettivo con una nuova che minimizzi la somma di tutte le variabili artificiali.
      3. +
    3. Fase 1: Risolvi il nuovo problema con il metodo Simplex.
      4. +
    4. Se il Simplex termina con ancora variabili artificiali nella base, allora il problema è .
      5. +
    5. Una volta che le variabili artificiali sono fuori base, elimina le loro colonne e la nuova funzione obiettivo.
      6. +
    6. Riporta il tableau in forma base compiendo operazioni per azzerare i coefficienti delle variabili di base nella funzione obiettivo.
      7. +
    7. Fase 2: Risolvi il tableau con il metodo Simplex.
      8. +
    +     +     + + +

    + Una versione semplificata di un problema nella quale si ignorano uno o più vincoli. +

    +     + +

    + Un rilassamento che permette di misurare di quanto i vincoli vengono violati. +

    +

    + I vincoli vengono aggiunti alla funzione obiettivo assieme a un moltiplicatore, solitamente rappresentato con {r`u_n`}. +

    + +

    + Il sistema: +

    + {r` + \begin{cases} + z = 3 x_1 + 5 x_2\\ + 2 x_1 + 3 x_2 \geq 12\\ + - x_1 + 3 x_2 \geq 3\\ + x_1 \geq 0\\ + x_2 \geq 0 + \end{cases} + `} +

    + diventa: +

    + {r` + \begin{cases} + z_{LR} = 3 x_1 + 5 x_2 + u_1 ( 12 - 2 x_1 - 3 x_2 ) + u_2 ( 3 + x_1 - 3 x_2 )\\ + x_1 \geq 0\\ + x_2 \geq 0 + \end{cases} + `} +     +     + +

    + Il sistema che massimizza/minimizza i moltiplicatori di rilassamento di un qualsiasi sistema, detto primale. +

    +

    + Si dimostra che la sua soluzione (se esiste) è uguale alla soluzione del problema primale. +

    +     + +

    + Possiamo trasporre il tableau e sostituire le variabili {r`x_n`} con variabili {r`u_n`} per ottenere il sistema duale! +

    +

    + I maggiori e minori dei vincoli diventeranno maggiori e minori delle variabili e viceversa. +

    +     + +
      +
    • Se un problema ha una soluzione finita, allora anche il suo duale la avrà.
      • +
    • Se un problema è , allora il suo duale potrà essere oppure .
      • +
    • Se un problema è , allora il suo duale sarà certamente .
      • +
    +     +     + + +

    + TODO: una complicata dimostrazione per dire varie cose. Probabilmente si riesce a saltare se non si dà l'orale... +

    +     + +

    + Il teorema che dimostra l'equivalenza tra primale e duale. +

    +

    + TODO: Anche qui c'è una lunga dimostrazione... +

    +     + +

    + Il teorema che dimostra che il valore della funzione obiettivo del duale (di un qualsiasi tableau) è sempre minore o uguale/maggiore o uguale alla soluzione del corrispettivo primale. +

    +

    + TODO: Dimostrazione cortina, ma sembra complicata.