0a6cap 20 Optimizacion, Funciones de Varias Variables

CAP TULO 20

Optimización: funciones de varias variables 20.1 20.2 20 .2 20.3 20. 3 20.4 20. 4 20.5 20 .5 20.6 20. 6

REPRESENTACIÓN REPRESENTACI ÓN GRÁFICA GRÁFICA DE FUNCIONES DE DOS VARIA ARIABLES BLES DERIV DERI VAD ADAS AS PAR PARCIA CIALES LES OPTIMIZACIÓN OPTIMIZACI ÓN DE LAS LAS FUNCIONES DE DOS VARIA VARIABLES BLES APLICACIONES APLIC ACIONES DE LA OPTIMIZACI OPTIMIZACIÓN ÓN DE DOS VARIA VARIABLES BLES OPT OP TIMIZ IMIZA ACI CIÓN ÓN DE DE  n VARIABLES (OPCIONAL) OPTIMIZACIÓN OPTIMIZACI ÓN SUJET SUJETA A A REST RESTRICCIONES RICCIONES (OPCIONAL) (OPCIONAL)

Términos y conceptos clave Fórmulas importantes Ejercicios adicionales Evaluación del capítulo Minicaso: Modelo de inventario de pedidos retrasados

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ESCENARIO DE MOTIVACIÓN: Método de los mínimos cuadrados, o cómo hallar la curva de mejor  ajuste para un conjunto de puntos de datos

A lo largo del texto se ha comentado la noción de estimación de relaciones matemáticas. En los capítulos 2, 5, 6 y 7 se vieron aplicaciones reales en las que se utilizaron puntos muestrales de datos para determinar las funciones de estimación lineales, cuadráticas y exponenciales. En cada caso, dichos puntos fueron seleccionados para uso en la determinación de las funciones de estimación. Dado un conjunto de puntos de datos y suponiendo una forma funcional (por ejemplo, lineal, cuadrática, etc.), el método de los mínimos cuadrados es uno de los más populares para determinar el “mejor” ajuste para los datos. En este capítulo se verá que el modelo de los mínimos cuadrados se basa en métodos de optimización (ejemplo 21).

En los capítulos 15 a 17 se ofreció una metodología para analizar las funciones que contienen una variable independiente. En las aplicaciones reales, un criterio u objetivo de decisión se basa a menudo en más de una variable. Como se mencionó en el capítulo 4, cuando las funciones incluyen más de una variable independiente se llaman funciones multivaria das o  funciones de varias variables. Se cuenta con métodos del cálculo diferencial para examinar su comportamiento y determinar los valores óptimos (máximos y mínimos). Al estudiar algunos de esos procedimientos en el presente capítulo, se verá que se parecen a los que se aplicaron a las funciones de una variable independiente. Este capítulo se concentrará inicialmente en las  funciones bivariadas (las que contienen dos variables independientes). Se describirán sus gráficas y luego se dará una explicación de las derivadas de esas funciones y su interpretación. A continuación se expondrán los métodos para obtener sus valores óptimos. Luego vendrá una sección en que se comentan las aplicaciones de las funciones bivariadas. La explicación abarcará la optimización de las funciones de n variables. El tema de la optimización restringida se aborda en la última sección del capítulo.

20.1

Representación gráfica de funciones de dos variables Representación gráfica Una función que incluye una variable dependiente  z y dos variables independientes  x  y  y puede representarse con la notación

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En general, las funciones no lineales que contienen una variable independiente se grafican como curvas en dos dimensiones. Y la gráfica de las funciones no lineales que contienen dos variables independientes son superficies curvas en tres dimensiones. Entre los ejemplos de superficies no lineales se encuentran la superficie ondulante de un campo de golf, la pendiente del terreno para esquiar y la vela de una embarcación de navegación. Un punto importante es que estas funciones se representan con superficies, no con sólidos.

Trazado de funciones de dos variables Aunque la graficación en tres dimensiones es difícil, se dispone de técnicas que pueden aplicarse en algunos casos para trazar la forma general de la gráfica de una función bivariada. El material que se explica a continuación se entenderá mejor si se conocen las gráficas de estas funciones. Considérese la función bivariada  z

f ( f ( x,  x, y)

25

x2

 y 2

(20.2)

donde 0 Յ x Յ 5 y 0 Յ y Յ 5. A fin de trazar esta función, se fijará el valor de una de las variables independientes y se graficará la función resultante. Por ejemplo, si se hace  y ϭ 0, la función f se convierte en

o bien

 z

25

x2

 z

25

x2

02

(20.3)

Al fijar el valor de una de las variables, la función se reformula en términos de la otra variable independiente. Es decir, una vez especificado el valor de la variable independiente, el de la variable dependiente varía con el valor de la variable independiente restante. Dada la ecuación (20.3), la tabla 20.1 indica algunos valores de  x y los valores resultantes de  z.

Tabla 20.1

 x   z

25

2

 x 

0 25

1 24

2 21

3 16

4 9

5 0

La figura 20.1 es una gráfica parcial de la función con el valor de  y fijado en 0. Si se hace y ϭ 0, la gráfica de la ecuación (20.3) debe estar en el plano  xz. Un estudio detenido

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 z = f ( f ( x, y) y) 30 25

–  x 2 – y  z = 25 – x

2

donde y = 0

20 15 10 5  x 1

2

3

4

5

1 2 3

Figura 20.1 Gráfica parcial de f ( x   x , y) = 25 Ϫ  x 2 Ϫ y2.

Tabla 20.2

 y

 z

25

y

2

0 25

4  y

1 24

5

2 21

3 16

4 9

5 0

 z = f ( f ( x, y) y) 30 25

– x 2 – y  z = 25 – x

20 15  x = 0

10 5

 y = 0

2

6

7

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 z = f ( f ( x, y) y) 30 25 – y  z = 25 – y

2

20 15 10 5

Figura 20.3 Traza con x ϭ 0 vista a lo largo del eje  x .

 y 5

4

3

2

1

Y si se examina atentamente la figura 20.2 en una dirección paralela al eje  x , se verá que la gráfica de esta ecuación es una parte de la parábola cóncava hacia abajo. La figura 20.3 contiene lo que se vería si se observara a lo largo del eje  x .

Definición: Traza Si z ϭ f ( x   x , y), una traza es la gráfica de  f cuando una variable se mantiene constante. Observe la figura 20.2. Las dos partes de  f que allí se muestran son trazas. Una es una traza cuando  y ϭ 0, en tanto que la otra es una traza donde  x  ϭ 0. Cada traza representa una costilla en la superficie que simboliza la función. La figura 20.4 presenta una gráfica de la función que incluye cuatro trazas adicionales. Haciendo y ϭ 1, la función se convierte en  f (  f ( x,  x, y)

25 24

x2 x2

12

La traza que representa a esta función es paralela al plano  xz y se encuentra una unidad fuera a lo largo del eje positivo  y. De manera análoga, si se hace  y ϭ 3, se tiene  f (  f ( x,  x, y)

25 16

x2 x2

32

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( x, y )  z = f ( f  x, 30

25

– x 2 – y 2  z = 25 – x

20

15

10  y = 0 5  y = 1

 x = 0

 x 1

 x = 1 1

2

3

4

5

6

7

 y = 3

2

 x = 3 3 4 5

Figura 20.2 Gráfica de

 f ( x   x , y) = 25 Ϫ x 2 Ϫ  y2.

 y

Por lo tanto, un procedimiento que puede en ocasiones ofrecer una gráfica aproximada de una función de la forma  z ϭ f ( x   x , y) consiste en suponer valores selectos de  x y y, para luego graficar las trazas que representan las funciones resultantes.

NOTA

Conviene tener presente en este momento una observación importante en relación con la gráfica de una función  f ( x  y  x  y). Siempre que

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1. 2. 3. 4. 5.

20.2

f ( f ( x,  x, y) f ( f ( x,  x, y) f ( f ( x,  x, y) f ( f ( x,  x, y) f ( f ( x,  x, y)

16 x 2  y 2, dond donde e0 x 4y0 y 4 9 x 2  y 2, dond donde e0 x 3y0 y 3 4 x 2  y 2, dond donde e0 x 2y0 y 2 25 x 2 /4 y 2 /4, donde 0 x 10 y 0 y x 2  y 2, dond donde e0 x 5y0 y 5

10

Derivadas parciales Aunque más complejo, el cálculo de las funciones bivariadas bivariadas se asemeja mucho al de las funciones de una sola variable. En la presente sección se hablará de las derivadas de estas funciones y de su interpretació interpretación. n.

Derivadas de funciones de dos variables En las funciones de una sola variable, la derivada representa la tasa instantánea de cambio en la variable dependiente respecto del que se opera en la variable independiente. En las funciones bivariadas se tienen dos derivadas parciales. Estas derivadas representan la tasa instantánea de cambio en la variable dependiente respecto de los cambios de las dos variables independientes, tomadas por separado. En una función  z ϭ  f ( x   x ,  y), puede calcularse una derivada parcial respecto de cada variable independiente. La derivada parcial tomada respecto de x se denota mediante  z  x

o

f  x

La derivada parcial tomada respecto de  y se indica mediante

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Por fortuna, las derivadas parciales se obtienen más fácilmente empleando las mismas reglas de diferenciación utilizadas en los capítulos 15 a 17. La única excepción es que, cuando se encuentra una derivada parcial respecto de una variable independiente, se supone que se mantiene constante a la otra. Por ejemplo, al calcular la derivada parcial respecto de  x , se supone que y es constante. Y un punto muy importante es que la variable que se supone constante debe tratarse como una constante al aplicar las reglas de diferenciación.

Ejemplo 3

Encuentre las derivadas parciales respecto de  f  x  y f  y para la función  f (  f ( x,  x, y)

5 x 2

6 y 3

SOLUCIÓN Primero, para determinar la derivada parcial respecto de x se supondrá que y es constante. Al diferenciar término por término, se observa que la derivada de 5 x 2 respecto de x es 10 x . Al diferenciar el segundo término, no se olvide que se supone que  y es constante. Así pues, este término presenta la forma general 6(constante)3 que es simplemente una constante. Una constante no cambia de valor al hacerlo otras variables (o, como se señaló en el capítulo 15, la derivada de una constante es 0), por lo cual la derivada del segundo término es 0. Por consiguiente,  f  x

10 x 10 x

0

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consiste en recordar cuál variable se supone que era constante. Cuando  y se mantiene constante, puede rearreglarse 4 xy para que tenga la forma Otro método

o bien

 f (  f ( x, y)

constante x

 f ( x,  x, y)

(4 y)  y) x  x

Al agrupar 4 y  y, este término presenta la forma general de una constante 4 y por x . Y la derivada de una constante multiplicada por x es la constante, o  f  x

4 y

Para calcular f  y, se supone que  x es constante. Al aplicar la regla del producto se obtiene o bien

 f  y

(0)( y)  y)

 f  y

4 x

(1)(4 x)  x)

O usando el otro procedimiento, el factor 4 x es constante (al mantener constante x ) y puede considerarse que f tiene la forma  f (  f ( x, y)

constante y (4 x)  x) y  y

La derivada respecto de y es la constante, o bien  f  y

Ejemplo 5

Encuentre f  x  y f  y si

4 x

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Ejercicio de práctica Verifique las expresiones para  f  x  y f  y calculándolas utilizando ahora la regla del producto.

Ejemplo 6

Calcule f  x  y f  y si e x

 f (  f ( x,  x, y)

2

 y 2

SOLUCIÓN Al aplicar las reglas de diferenciación de las funciones exponenciales,  f  x

(2 x

0) e  e x

2

 y 2

2 xe x

2

 y 2

 f  y

(0

2 y)  y) e  e x

2

 y 2

2 ye x

2

 y 2

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II

 f  y es una expresión general para la pendiente de la tangente de la familia de trazas paralelas al plano yz.

La derivada parcial  f  x  estima el cambio en z cuando se da un cambio en x , suponiendo que  y se mantenga constante. En la sección 20.1 se vio que, cuando  y es mantenida constante, las trazas correspondientes se grafican paralelas al plano xz. La pendiente de esas trazas la representa  f  x . De manera semejante, f  y supone que x se conserva constante. Cuando se mantuvo constante x en la sección 20.1, el resultado fue una familia de trazas paralelas al plano  yz. Y  f  y representa la pendiente de esas trazas. La figura 20.5 muestra la representación de la pendiente. La otra interpretación de las derivadas parciales es la de la tasa instantánea de cambio. Como en el caso de las funciones de una sola variable, las derivadas parciales pueden emplearse para aproximar los cambios de valor de la variable dependiente, si se produce un

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cambio en una de las variables independientes. Por ejemplo, f  x  puede servir para aproximar el cambio de f ( x   x , y), cuando se da un cambio en  x y se supone que  y es constante. La derivada parcial  f  y puede utilizarse para aproximar el cambio de  f ( x   x ,  y), dado un cambio en  y, suponiendo que x es constante. El siguiente ejemplo ilustra esta interpretación.

Ejemplo 8

(Interrelaciones de la demanda de varios productos) Hasta ahora se ha supuesto que la demanda de un producto depende, exclusivamente, de su precio. Así, las funciones de demanda analizadas presentan la forma q

ϭ  f ( p)

Con frecuencia, la demanda de un producto o servicio recibe el influjo no sólo de su precio, sino también de los precios de otros productos o servicios. La ecuación (20.5) es una función de demanda que expresa la cantidad demandada del producto 1 (q1) en términos de su precio ( p1) y también de los precios de otros dos productos ( p  p2 y p3), todos ellos expresados en dólares. q

f ( f ( p

p

p )

10 00 000

2 5 p

3 p

1.5 p

(20.5)

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Ejercicio de práctica En la función de demanda q1

f ( f ( p1 , p2 , p3 )

120 00 000

0.5 p 21

0.4 p 22

0.2 p 23

a)

Calcule todas las derivadas parciales. b) Si los precios actuales de los tres productos son p1 = 10, p2 = 20 y p3 = 30, evalúe las derivadas parciales e interprete su significado. c) ¿Qué sugerirán la función de demanda y sus derivadas parciales respecto de la interdependencia entre los tres productos?  Respuesta: a) f  p ϭ Ϫ p1, f  p ϭ Ϫ0.8 p2, f  p ϭ Ϫ0.4 p3; b) f  p (10, 1

Ejemplo 9

1

2

3

20, 30) = –10,  f  p (10, 20, 30) ϭ Ϫ16, f  p (10, 20, 30) ϭ Ϫ12; c) son productos complementarios. 2

3

(Gastos de publicidad) Un fabricante nacional estima que el número de unidades que vende cada año es una función de los gastos hechos en la publicidad por radio y televisión. La función que especifica esta relación es

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 f (41,  f (41, 20)

f ( f (40, 20) 281699 2816990 0 276800 2768000 0 48 990 unidades unidades

La diferencia entre los incrementos real y estimado usando  f  x  es de de 48 990 Ϫ 49 00 0000 ϭ Ϫ10 unidades. El signo de menos indica que la derivada parcial sobreestimó el cambio verdadero. Supóngase ahora que se quiere quiere determinar el efecto, si se destinan $1 000 más a la publicidad por radio y no a la publicidad por televisión. La derivada parcial tomada con respecto de y aproximará este cambio:  f  y

40 00 000

40 y

10 x

Al evaluar f  y cuando x ϭ 40 y cuando y ϭ 20, se obtiene  f  y(40 (40, 20) 20)

40000 0000 40 00 000

40(20 (20) 10(4 10(40) 0) 800 400 38 80 800

Así pues, un aumento de $1 000 en los gastos de publicidad por radio originará un incremento aprogastos de este tipo de publicidad, las  ximado de 38 800 unidades. Con un aumento de $1 000 en los gastos

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Derivadas parciales de primer orden

f  x

Derivadas parciales de segundo orden

diferenciar respecto de x de x

 f  xx

(derivada parcial pura de segundo orden)

diferenciar respecto de y de y

f  xy

(derivada parcial mixta)

f 

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b)

62.

Haciendo uso de las derivadas derivadas parciales, estime el cambio esperado en f ( x   x ,  y), si  x  aumenta en una unidad. Comparee el cambio cambio real con con el cambio cambio estimado. estimado. c) Compar Repita ita los inc inciso isoss b) y c), suponiendo un posible incremento de una unidad en  y. d ) Rep 3 En f ( x   x , y) ϭ 20 x  Ϫ 30 y3 ϩ 10 x 2 y: Dete term rmin inee f (20, 10). a) De derivadas parciales, estime el cambio esperado en f ( x  y b) Haciendo uso de las derivadas  x   y), si  x  aumenta

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Dichos puntos tienen en dos dimensiones el mismo significado que en tres.

Definición: Máximo relativo Se dice que una función  z ϭ f ( x   x , y) tiene un  máximo relativo cuando x ϭ a y y ϭ b si para todos los puntos ( x   x , y) “suficientemente cercanos” a ( a, b),

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Condición necesaria de extremos relativos Una condición necesaria para la existencia de un máximo relativo o un mínimo relativo de una función f cuyas derivadas parciales  f  x  y f  y existen, establece que  f  x

0

y

f  y

0

(20.6)

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4

 j

( y j

y p )2

[ y j

(ax j

  j

1 4

 j

b)] 2

1

Pongamos el simple caso de una compañía que ha reunido tres niveles diferentes de precios. La tabla 20.3 contiene las combinaciones de precio-demanda. La figura 20.21 es una gráfica de los datos. Supóngase que se desea determinar la línea del mejor ajuste para esos puntos de datos, usando para ello el modelo de mínimos cuadrados. La función de mínimos cuadrados se genera mediante la ecuación (20.23). S

f ( f (a, b) 3

[ y j  j

(5a

(demanda en miles de unidades)  x  (precio en dólares)  y

b)] 2

1

[50

Tabla 20.3

(ax j

b)] 2

(10a

[30

50 5

b)] 2

[20

30 10

(15a

20 15

 y

70

  s   e    d   a 60    d    i   n   u   e    d 50   s   e    l    i   m40   n   e  ,   a    d 30   n   a   m   e    D20

(5, 50)

(10, 30) (15, 20)

b)] 2

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500 700a 700a  f b

50a 60b 60b

10b 10b 1700

200a

600

20b 20b

2[50 (5a b)]( 1) 2[30 (10a 2[20 (15a b)]( 1) 100 10a 2b 60 20a 2b 60a 60a 6b 200

600

b)](

1)

40

30a

450a

30b 30b

2b

Si hacemos iguales a 0 estas dos derivadas, resultarán las dos ecuaciones siguientes: 700a 700a

60b 60b

1700

(20.24)

60a 60a

6b

200

(20.25)

La multiplicación de la ecuación (20.25) por Ϫ10 y la adición del producto a la ecuación (20.24) dan 700a 700a 600a 600a

60b 60b 60b 60b

1700 2000

100a 100a

300 a

3

Al sustituir a ϭϪ3 en la ecuación (20.25) se obtiene 60( 3) 3)

6b

200

6b

380

b

63 13

Para verificar que el punto crítico produce un valor mínimo de S ,

 D(  D (

 f aa

700

f ab

60

 f bb

6

f ba

60

3, 63 13 )

(700)(6) (60)2 4 2 00 00 3 6 00 00 600 0

Puesto que  D Ͼ 0 y tanto  f aa como  f bb son positivas, puede llegarse a la conclusión de que la suma de los cuadrados de las desviaciones S se minimiza cuando a ϭ Ϫ3 y b ϭ 63 13, o cuando los puntos de datos se ajustan con una línea recta teniendo una pendiente de Ϫ3 y una intersección de 63 13 con el eje y. La ecuación de esta línea es

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NOTA

Este ejemplo está dirigido para ilustrar los fundamentos para esta popular técnica de estimación. Afortunadamente, la puesta en práctica (uso) real del método de los mínimos cuadrados no requiere de la formulación de la función de la suma de los cuadrados y análisis de optimización como se demostró en este ejemplo. Por lo regular, el análisis de mínimos cuadrados se realiza al introducir los puntos de datos de muestra en una calculadora portátil o bien en cualquiera de los paquetes de programas estadísticos que se encuentran disponibles en una gran variedad en el mercado.

Secci Sec ción ón 20. 20.4 4 Ej Ejer erci cici cios os de segu seguim imie ient nto o 1.

Un fabricante estima que las ventas anuales (en unidades) son una función de los gastos hechos en la publicidad por radio y televisión. La función que especifica la relación es  z

40000 x 40000 x

60000 y 60000 y

5 x 2

10 y 2

10 xy

donde z es el número de unidades vendidas cada año, la  x  denota la cantidad destinada a la publicidad por televisión y la  y representa la que se dedica a la publicidad por radio (tanto x como  y se dan en miles de dólares). radio y televisión a fin de maximizar a) Determine cuánto debería gastarse en la publicidad por radio el número de unidades vendidas. espera que sea sea el número máximo máximo de unidades unidades?? b) ¿Cuál se espera 2. Una compañía vende dos productos. Se estima que el ingreso total conseguido con ellos es una función del número de unidades vendidas. En concreto, la función es  R

3.

30000 x 30000 x

15000 y 15000 y

10 x 2

10 y 2

10 xy

donde  R es el ingreso total y tanto  x  como  y indican los números de unidades vendidas de ambos productos. a) ¿Cuántas unidades de cada producto deberían fabricarse con objeto de maximizar el ingreso total? ¿Cuál ál es el ingres ingresoo máximo? máximo? b) ¿Cu Una empresa vende dos productos, Sus funciones de demanda son q1

110

q2

90

4 p1 2 p1

p2 3 p2

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 y (0, 40)

(30, 10)  x

(–30, –10)

Figura 20.21 Localización de las tiendas de departamentos. 5.

6.

Localización de aeropuertos. Se planea un nuevo aeropuerto que dará servicio a cuatro áreas metropolitanas. Las localizaciones relativas de éstas en un conjunto de ejes coordenados son (20, 5), (0, 30), (Ϫ30, Ϫ10) y (Ϫ5, Ϫ5), donde las coordenadas se expresan en millas. La figura 20.23 indica las localizaciones relativas de las cuatro ciudades. Determine la ubicación del aeropuerto ( x   x , y) que minimice la suma de los cuadrados de las distancias entre el aeropuerto y cada área metropolitana. En el ejemplo 20, suponga que el número de miembros de la organización para la conservación de la salud que viven viven en tres municipios municipios es 20 000, 10 000 y 30 000, respectiv respectivament amente, e, en los municipios A, B y C . Suponga además que la organización desea conocer la localización ( x , y) que minimiza la suma de los productos del número de miembros de cada municipio y el cuadrado de la distancia que separa las ciudades y la clínica.  y

(0, 30)

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Este objetivo puede formularse como 3

minimice

n j d j2  j

7.

8.

Tabla 20.4

20.5

1

donde n j es el número de miembros que viven en la ciudad j y d  j denota la distancia entre el municipio j y la clínica. Determine la localización de la clínica. Con los puntos de datos (2, 2), (Ϫ3, 17) y (10, Ϫ22), determine la ecuación de la línea del me jor ajuste utilizando el modelo de mínimos cuadrados. Con los puntos de datos relativos a la relación entre precio y demanda de la tabla 20.4, determine la ecuación de la línea del mejor ajuste a esos puntos de datos empleando el modelo de mínimos cuadrados.

(demanda en miles de unidades)  x  (precio en dólares)  y

200 30

160 40

120 50

Optimización de  n variables (opcional) Cuando una función contiene más de dos variables independientes, el proceso con que se identifican los máximos y mínimos relativos se parece mucho al que se aplica a funciones con dos variables independientes. Antes de explicar el proceso, definiremos esos extremos relativos.

Definición: Máximo relativo Se dice que una función  y ϭ f ( x   x 1, x 2,..., x n) tiene un  máximo relativo en x 1 ϭ a1,  x 2 ϭ a2,..., x n = an, si para todos los puntos ( x 1, x 2,..., x n), suficientemente cercanos a (a1, a2,..., an),  f (a1, a2,..., an) Ն f ( x   x 1, x 2,..., x n)

Definición: Mínimo relativo Se dice que una función  y ϭ f ( x   x 1, x 2,..., x n) tiene un  mínimo relativo en  x 1 ϭ a1,

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Condición necesaria para los extremos relativos Una condición necesaria para un máximo relativo o un mínimo relativo de una función cuyas deriva derivadas das parciales  f  x 1,  f  x 2,...,  f  x  existan es n

 f  x

0, f  x

1

0,..., f  x

2

(20.26)

0

n

La condición necesaria exige que todas las primeras derivadas parciales de  f sean iguales a 0.

Ejemplo 22

Para localizar los candidatos a convertirse en puntos extremos relativos en  f (  f ( x  x1 , x2 , x3 )

x 12

2 x1 x2

2 x 22

2 x1 x3

se calculan las primeras derivadas parciales  f  x

1

 f  x

2

 f  x

3

2 x1 2 x1 2 x1

2 x2

2 x3

4 x2 8 x3

2

Puesto que las tres derivadas deben ser 0, las tres ecuaciones 2 x1

2 x2

2 x1

4 x2

2 x1

2 x3

0 0

8 x3

2

4 x 23

2 x3

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En la función f ( x   x 1, x 2, x 3,..., x n), la matriz hessiana es cuadrada y con dimensión ( n ϫ n). La diagonal principal está constituida por las segundas derivadas parciales puras y los elementos no diagonales son derivadas parciales mixtas. La matriz es también simétrica si métrica alrededor de la diagonal principal cuando las segundas derivadas parciales son continuas. En tales circunstancias, las derivadas parciales parciales mixtas que se toman respecto de las dos mismas variables son iguales. Es decir,  f  x  f  x  ϭ f  x  x . Para una matriz hessiana ( n ϫ n), puede identificarse un conjunto de n sub submat matric rices es. La primera de ellas es la submatriz (1 ϫ 1) formada por el elemento situado en el renglón 1 y en la columna 1,  f  x 1 x 1. Denotemos esta matriz como H1, donde i

 j

 j

i

( f  x  x )

H1

La segunda submatriz es la matriz (2

ϫ

H3

ϫ

1

2)  f  x  x  f  x  x

H2

La tercera submatriz es la matriz (3

1

1

1

2

1

 f  x  x  f  x  x 1

2

2

2

3)  f  x  x  f  x  x  f  x  x 1

1

2

1

3

1

 f  x  x  f  x  x  f  x  x 1

2

2

2

3

2

 f  x  x  f  x  x  f  x  x 1

3

2

3

3

3

La n-ésima submatriz es la matriz hessiana propiamente dicha, o sea H n ϭ H. Los determinantes de estas submatrices reciben el nombre de  menores principales. El menor principal asociado a la i-ésima submatriz puede denotarse como ⌬i.

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Ejemplo 23

Continuando ahora con el ejemplo 22, la matriz hessiana será 2 2 2

H

2 4 0

2 0 8

Las submatrices y los valores correspondientes de los menores principales son H1

(2)

H2

2 2

2 4

H3

2 2 2

2 4 0

2 0 8

1

2

2

4

3

16

Dado que los menores principales ⌬1, ⌬2 y ⌬3 son positivos, se extrae la conclusión de que se presenta un mínimo relativo en el punto crítico (Ϫ1, Ϫ 12, 12, Ϫ12).

Ejemplo 24

Localice cualquier punto crítico y determine su naturaleza en la función 2 x 31

 f (  f ( x  x1 , x 2 , x 3 )

6 x1 x3

x 22

2 x2

6 x 32

5

SOLUCIÓN Las primeras derivadas parciales se calculan y se hacen iguales a 0, de la manera siguiente:  f  x

6 x 12

6 x3

0

 f  x

2

2 x2

0

1

2

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donde C es el costo total (en dólares) de producir q1, q2 y q3 unidades de los productos 1, 2 y 3, respectivamente. a) Determine las cantidades que producirán un mínimo costo total. Confirme que el punto crítico sea un mínimo relativo. b) ¿Cuál es el mínimo mínimo costo total total esperado esperado??

20.6

Optimización sujeta a restricciones (opcional) Nuestro análisis de los métodos de optimización basados en el cálculo se centró en la  op timización no restringida. En muchas aplicaciones del modelado matemático interviene la optimización de una  función objetivo sujeta a ciertas condiciones restrictivas, o simplemente  restricciones. Estas restricciones representan limitaciones capaces de influir en el grado que se optimizan las funciones objetivo. Y pueden reflejar limitaciones como escasez de recursos (por ejemplo, mano de obra, materiales o capital), poca demanda de productos, metas de ventas, etc. Los problemas que ofrece esta estructura se consideran  problemas de optimización restringida. Se estudió un subconjunto de ellos al examinar la programación lineal (capítulos 10 a 12). En esta sección nos ocuparemos de un método con que se resuelven ciertos problemas de optimización no lineal restringida.

Método del multiplicador de Lagrange (restricción de la igualdad) Considérese el problema de optimización restringida Máximo (o mínimo) y f ( x  x1 , x2 ) sujeto a g ( x  x1 , x2 ) k

(20.27)

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Condiciones necesarias para los extremos relativos  L x  ϭ 0 1

(20.29)

 L x  ϭ 0 2

 Lλ  ϭ 0

Ejemplo 25

Considere el problema Máximo f ( f ( x  x1 , x2 )

25

sujeto a 2 x 2 x1

4

x2

x 21

 x 22

Con el método del multiplicador de Lagrange se transforma este problema en la forma no restringida  L(  L(

)

2

2

(2

4)

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Si este valor de  x 1 se sustituye en la ecuación (20.32), 2(2 x2 )

x2

4

0

5 x2  x2

4 0.8

4 5

Si este valor se sustituye en la ecuación (20.34), x 1 ϭ 1.6. Por otra parte, la sustitución de x 2 ϭ 0.8 en la ecuación (20.31) da λ  ϭ Ϫ1.6. Por lo tanto,  x 1 ϭ 1.6,  x 2 ϭ 0.8 y λ  ϭ Ϫ1.6 son valores críticos en la función lagrangiana. Estos valores de x 1 y x 2 también representan los únicos puntos candidatos pa❑ ra un máximo (o mínimo) relativo.

Condición suficiente Para estimar el comportamiento de  L( x   x 1,  x 2, λ ) en cualquier valor crítico, deberá determinarse la  matriz hessiana acotada H B, donde 0

 g x

1

 g x

2

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 f (  f ( x  x 1, x 2)

35  B

30

25 2 x1 + x2 = 4

(1.6, 0.8, 21.8)  N   f (  f  x ( x 1, x 2) = 25 – x – x 12 – x 22

20  A 15

10

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NOTA

La estructura de este problema se parece mucho a la del ejemplo 11 de la página 834 del capítulo 17. La ecuación (17.12) es la función objetivo; la ecuación (17.1) es una restricción. Ese problema fue resuelto al despejar una variable en términos de la otra en la ecuación (17.13) y al sustituirla en la función objetivo. Este procedimiento también puede aplicarse al ejemplo 25. ¿Por qué, entonces, se recurre al método del multiplicador de Lagrange? El ejemplo 25 es un problema relativamente simple. La estructura de la restricción o restricciones de un problema a menudo no permiten las sustituciones; ¡y es allí donde entra el multiplicador de Lagrange!

Caso de restricción de una sola igualdad con  n variables En un problema de la forma Máximo (o mínimo)

f (

)

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H B

2

0  g x  g x

 g x  L x  x  L x  x

0  g x  g x  g x

 g x  L x  x  L x  x  L x  x

1 2

H B

1

3

2 3

 g x  L x  x  L x  x

1

2

1

1

2

1

2

2

2

 g x  L x  x  L x  x  L x  x

1

1

1

 g x  L x  x  L x  x  L x  x

2

1

2

1

3

1

3

1

2

2

2

3

2

1

n

H B

2

3

3

3

0 g x  g x  g x  g x  L x  x  L x  x  L x  x g x  L x  x  L x  x  L x  x ..........................  g x  L x  x  L x  x  L x  x 1

H B

3

2

n

1

1

1

1

2

1

n

2

2

1

2

2

2

n

n

n

1

n

2

n

n

Los menores principales para estas submatrices pueden denotarse como

⌬ B

, ⌬ B3,... ⌬ Bn.

2

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 L x

5 x1 x3

2

0

 L x

5 x1 x2

3

0

 x1

2 x2

2

3

 L x

3

3 x3

24

0

Estas cuatro ecuaciones pueden reescribirse como 5 x2 x3

(20.39)

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H B

Los menores principales acotados son

0 1 2 3

1 0 40 3

20

2 40 3

0 40

3 20 40 0

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Extensiones El método de Lagrange puede ampliarse al caso de restricciones múltiples y al de conjuntos de restricciones que comprenden tipos de restricción de desigualdad e igualdad. Sin embargo, esas situaciones rebasan el alcance de este libro.

Secci Sec ción ón 20. 20.6 6 Ej Ejer erci cici cios os de segu seguim imie ient nto o

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x2

3 y 2

37.

f ( f ( x,  x, y)

38.

f ( f ( x,  x, y)

39.

f ( f ( x,  x, y)

2 x 3

 y 3

3 x 2

40.

f ( f ( x,  x, y)

xy

 x 1/ 

 y 1/ 

41.

f ( f ( x,  x, y)

6 x 2

42.

f ( f (

)

4 x 2

4 x

4ln

10

3 y 2

12 x

30 x

9 y

1.5 y 2

y2 2

36 y

5 12 x 12 x

6 y 2

10,

0

90 y

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