1. ## Interesting optimization problem

Hi,

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

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

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

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

3. I took a stab at this problem, and find the optimum to arise when $\displaystyle a_j = \lambda P_j$ and $\displaystyle \sum a_j = \lambda$. Since $\displaystyle \lambda$ is an arbitrary scalar, simply take $\displaystyle \lambda = 1$ and you arrive at Boltzmann's equation for H.

Thus $\displaystyle J_{max} = \sum_j P_jlnP_j$
with $\displaystyle \sum_j P_j = 1$

4. 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 $\displaystyle {a_k}ln(P_k)$ Where $\displaystyle P_k$ 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.