Do not know how to approach this problem. The task is to convert the problem of mathematical programming:
$\max(\prod_{i=1}^nx_i)$
$\sum_{i=1}^na_ix_i\leq c$
$c \gt 0, a_i \gt 0$
into a problem of dynamic programming. Write down Bellman equations for this problem and find an optimal solution when
$n=3, a_1 = 1, a_2 = 2, a_3 = 3, c = 10$
I do not ask for a total solution, but I just need conceptual hints. By now I've tried $\lambda-method$ - used natural log of the value function, so that it transformed from a product to a sum of logs, then linearized them on their intervals, but all that resulted in a terrible linear programming problem, which I still didn't know how to convert to a dynamic programming problem.
Think of choosing the $x_i$'s sequentially, i.e., $x_1$ is chosen first, $x_2$ is chosen second, and followed by $x_3$. To make things simpler, let's assume that the $x_i$'s can only take integer values (otherwise we would have to discretize the continuous space).
In order to capture the constraint, at each stage we have a budget $B$, which takes values in $\{0, 1, \ldots, 10\}$ and for each of these values we compute the Value Function. Intuitively, for stage 3, think of budget $B$ as $(c - a_1 x_1 - a_2 x_2)$. Since the objective is increasing in $x_3$, $$V_3(B) = \begin{cases} \lfloor B/a_3 \rfloor\quad &\text{if}\ B \geq 0,\\ 0 &\text{otherwise.} \end{cases} $$ For stage 2, calculate $$ V_2(B) = \max_{0\leq x_2 \leq B}\{x_2 V_3(B - a_2 x_2)\}, $$ for all values of $B \in \{0, 1, \ldots, 10\}$.
Finally for stage 1, we can can calculate, $$ V_1(10) = \max_{0\leq x_1 \leq 10}\{x_1 V_2(10 - a_1 x_1)\}. $$