Interesting optimization problem

**luispipe** · May 28th 2010, 12:50 PM

Hi,

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

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

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

Thank you for your comments.

**GeoC** · May 31st 2010, 07:08 AM

Looks remarkably similar to Boltzman's characterization for the distribution of the elements of a system across states $j$ , where $P_j$ is the probability of occupancy of state $j$ . This is subject to the constraint that $\sum P_j = 1$ . In the case of Boltzmann, $a_j = P_j$

**GeoC** · May 31st 2010, 09:28 AM

I took a stab at this problem, and find the optimum to arise when $a_j = \lambda P_j$ 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 $\sum_j P_j = 1$

**SpringFan25** · May 31st 2010, 12:09 PM

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.

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