# is there a closed form solution to this continuous optimization problem?

Consider the function

\begin{eqnarray} \max_{t_1,\ldots,t_p \ge 0} V(p) & = & \sum_{i=1}^p [- \alpha t_i - \beta e^{\rho - \delta^{i-1}\theta} \prod_{k=1}^i t_k^{-\delta^{i-k}\Omega}]. \end{eqnarray}

Decision variables are $t_1,\ldots,t_p$; e is the natural log base; everything else is a strictly positive parameter.

Observations

1. It is possible to write $\ln(t_p)$ as a linear function of $\ln(t_1),\ldots,\ln(t_{p-1})$.

2. The function V(p) is concave.

3. Since the function is concave, it is easy to solve numerically, for example using Maple's NLPSolver.

Question: Is there either (a) a closed form expression for $t_1,\ldots,t_p$ as a function of the parameters, or (b) a simple recursion between $t_i$ and $t_{i+1}$, for $1 \le i \le p-1$?

Thank you for any suggestions.

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You said "function" but wrote an equation. What is the function? – Antonio Vargas Mar 25 '12 at 17:21
Thanks. V is a function of t_1,...t_p. The optimization is over these variables. – Nick Mar 28 '12 at 18:25