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Math Help - Interesting optimization problem

  1. #1
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    Interesting optimization problem

    Hi,

    I would like to know if the following optimization problem has any application (or has been used to solve anything):

    <br />
  \begin{array}{*{20}{c}}<br />
   {{\text{maximize}}} & {J=\sum\limits_{j = 1}^n {{a_j}ln {p_j} } } & {}  \\<br />
   {{\text{subject to}}} & {\sum\limits_{j = 1}^n {{p_j} = 1} } & {}  \\<br />
   {} & {{p_i} \geqslant 0,} & {i = 1, \ldots ,n}  \\<br />
 \end{array}<br />

    where a_i is a positive constant for i=1,...,n

    Thank you for your comments.
    Last edited by luispipe; May 28th 2010 at 01:21 PM.
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  2. #2
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    Looks remarkably similar to Boltzman's characterization for the distribution of the elements of a system across states j, where P_jis the probability of occupancy of state j. This is subject to the constraint that \sum P_j = 1. In the case of Boltzmann, a_j = P_j
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  3. #3
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    I took a stab at this problem, and find the optimum to arise when a_j = <br />
\lambda P_j and \sum a_j = \lambda. Since \lambda is an arbitrary scalar, simply take \lambda = 1 and you arrive at Boltzmann's equation for H.

    Thus J_{max} = \sum_j P_jlnP_j
    with \sum_j P_j = 1
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  4. #4
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    Certain economics problems could be expressed in this way.


    If a person must spread a fixed resource (say, Time), over J activities, and the Utility derived from each activity is equal to {a_k}ln(P_k) Where P_k is the proportion of your time that you spend doing activity k. You would maximise




    The constraints would come from:
    All time must be allocated
    All time must be positive

    It would be stupid to assume anyone had this utility function of course, but that never stopped economists before.
    Last edited by SpringFan25; May 31st 2010 at 02:22 PM.
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