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    1.1. Questions: D N Y DD NN YY

    Uploaded by

    Everton Colling
    This document provides an overview of process modeling and optimization techniques covered in a summer semester tutorial. It includes examples of using numerical optimization methods like the Golden Cut Interval Reduction method and Newton's method to find the minimum of a scalar function and the negative root of a polynomial. While numerical methods can determine local minima/roots, other techniques may be needed to find global minima since the example function is non-convex over its full domain.

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    © All Rights Reserved
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    1.1. Questions: D N Y DD NN YY

    Uploaded by

    Everton Colling
    14 views2 pages
    This document provides an overview of process modeling and optimization techniques covered in a summer semester tutorial. It includes examples of using numerical optimization methods like the Golden Cut Interval Reduction method and Newton's method to find the minimum of a scalar function and the negative root of a polynomial. While numerical methods can determine local minima/roots, other techniques may be needed to find global minima since the example function is non-convex over its full domain.

    Original Description:

    Tutorial Process Optimization

    Original Title

    PO SS2012 Tutorial01

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    This document provides an overview of process modeling and optimization techniques covered in a summer semester tutorial. It includes examples of using numerical optimization methods like the Golden Cut Interval Reduction method and Newton's method to find the minimum of a scalar function and the negative root of a polynomial. While numerical methods can determine local minima/roots, other techniques may be needed to find global minima since the example function is non-convex over its full domain.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    14 views2 pages

    1.1. Questions: D N Y DD NN YY

    Uploaded by

    Everton Colling
    This document provides an overview of process modeling and optimization techniques covered in a summer semester tutorial. It includes examples of using numerical optimization methods like the Golden Cut Interval Reduction method and Newton's method to find the minimum of a scalar function and the negative root of a polynomial. While numerical methods can determine local minima/roots, other techniques may be needed to find global minima since the example function is non-convex over its full domain.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    Process Modelling and Optimisation

    Tutorial 1

    Summer Semester 2012


    Department of Chemical Engineering
    Malte Behrens, Sergio Lucia
    Chair of Process Dynamics and Operations,
    Prof. Sebastian Engell

    1.1. Questions
    Describe different classes of optimization problem.

    1.2. Scalar Optimization


    The following optimisation problem is given:
    1 5
    4
    min
    x x 4 x 3 + 8x 2
    x[ 3, 5 ] 5
    3
    a. Determine a minimum numerically.
    Hint: To evaluate the function for given x values, first define the function in Matlabs
    command window by typing:
    f=inline('1/5*x.^5 - x.^4 - 4/3*x.^3 +8*x.^2')
    Now you can evaluate the functions easily, e.g.: f(5) ans = 33.33 )
    Apply 3 steps of the Golden Cut Interval Reduction method (t=0.61).
    Start with the interval [-2.0, 5.0].
    Interval Reduction algorithm (slightly different than in lecture):

    x1(0 ) = a (0 ) + (1 t ) b (0 ) a (0 )
    for

    x 2(0 ) = b (0 ) (1 t ) b (0 ) a (0 )

    i = 0,1,2, do

    ( ) ( )

    f x2(i ) f x1(i )

    if
    a

    (i +1)

    =a

    b (i +1) = x2(i )
    ( i +1 )

    x2

    = x1(i )

    a
    -2.0

    (i +1)

    = x1

    (i )

    b (i +1) = b (i )

    x1(i +1) = a (i +1) + (1 t ) b ( i +1) a (i +1)

    i
    0
    1
    2
    3

    ( ) ( )

    f x2(i ) < f x1(i )

    if

    (i )

    b
    5.0

    x1(i +1) = x2(i )

    x2(i +1) = b (i +1) (1 t ) b ( i +1) a (i +1)

    x1

    x2

    f(x1)

    f(x2)

    f(x2)> f(x1)

    Then use 3 steps of the Newton method, start from x=0.5.

    b. Use the necessary and the sufficient condition of a minimum to determine the global
    minimum analytically. And also analyse the limits of the feasible region.
    c. Plot the function f using: x=-3:0.1:5; plot(x,f(x)). Why cant you find the global
    minimum using the given numerical methods?

    1/2

    Process Modelling and Optimisation

    Tutorial 1

    1.3. Problem: Determining the roots of a polynomial by


    optimization
    Given is the following polynomial:
    P( x) = x 3 4.4 x 2 9.1x + 21.4

    It is known that only one of the roots of this polynomial is negative.


    a.

    Use the enclosure of the minimum procedure to bound the negative root. Take step size
    s =1

    b.

    Taking golden ratio t = 0.618 determine the value of the negative root of the given
    polynomial using the interval reduction method. The desired decimal accuracy is

    = 10 3 .
    c.

    How is it possible to numerically determine the other roots of the polynomial?

    2/2

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