Results 1 to 3 of 3

Math Help - Maximize the minimum value in a vector

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    5

    Maximize the minimum value in a vector

    Here's a problem I am trying to solve.

    Maximize the minimum value of I:

    I = C + R * P (* indicates coordinate-wise product)

    where I, C, R and P are n x 1 vectors with the condition that the values of R sum to 1 and all its values be in [0,1]. C and P would be given.

    Here is an example with n = 2 (I am writing these vectors transposed so I can type it easily). C is given as [100 120]. P is given as [50 40]

    [I1 I2] = [100 120] + [R1 R2] * [50 40]

    With n = 2, this breaks down to a system of 2 equations that I was able to solve:

    I1 = 100 + (1 - R2) (50)
    I2 = 120 + R2 (40)

    I set I1 = I2 since this would maximize the minimum value between I1 and I2 (however even this approach would break down under certain conditions).

    100 + 50 - 50R2 = 120 + 40R2
    150 - 50R2 = 120 + 40R2
    30 = 90R2
    R2 = 1/3, R1 = 2/3


    Is there any way to generally solve this problem? Algorithmic solutions would be helpful as well.
    Last edited by daaaaave; August 5th 2010 at 05:18 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,365
    Thanks
    1311
    Quote Originally Posted by daaaaave View Post
    Here's a problem I am trying to solve.

    Maximize the minimum value of I:

    I = C + R * P (* indicates dot product)

    where I, C, R and P are n x 1 vectors with the condition that the values of R sum to 1 and all its values be in [0,1]. C and P would be given.
    Are you sure that is what the problem says? R*P is a number while C is a vector. The sum C+ R*P is not defined.

    Or do you mean I= (C+ R)*P?

    Here is an example with n = 2 (I am writing these vectors transposed so I can type it easily). C is given as [100 120]. P is given as [50 40]

    [I1 I2] = [100 120] + [R1 R2] * [50 40]

    With n = 2, this breaks down to a system of 2 equations that I was able to solve:

    I1 = 100 + (1 - R2) (50)
    I2 = 120 + R2 (40)
    This product is NOT "dot product". You appear to be using [A, B]*[X, Y]= [AX, BY], a "coordinate wise" product.

    I set I1 = I2 since this would maximize the minimum value between I1 and I2 (however even this approach would break down under certain conditions).

    100 + 50 - 50R2 = 120 + 40R2
    150 - 50R2 = 120 + 40R2
    30 = 90R2
    R2 = 1/3, R1 = 2/3


    Is there any way to generally solve this problem? Algorithmic solutions would be helpful as well.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    Posts
    5
    Yes, I am sorry. I am talking about a coordinate-wise product, not a dot product.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: August 21st 2011, 12:12 PM
  2. Finding a minimum for a "vector" function
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: May 4th 2011, 06:57 AM
  3. Replies: 1
    Last Post: October 10th 2009, 10:16 PM
  4. Vector calc minimum problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 21st 2009, 11:18 PM
  5. [SOLVED] maximize minimum-> 2 functions, 2 variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 17th 2007, 12:34 PM

Search Tags


/mathhelpforum @mathhelpforum