Knapsack problem with diminishing prices I wonder if the knapsack problem with diminishing prices is already studied?
The problem is similar to the regular knapsack problem, except the price of each item is a decreasing function of the total size of the already selected items.

Given $n$ items each with size $s_i$ and price $v_i(s)$, where $s$ is the total size of the already selected items, a size limit $S$ and a total price $V$, we want to find a sequence of items $\langle e_1,\cdots,e_k \rangle$ so that $\sum_{i=1}^k s_{e_i} \le S$ and $\sum_{i=1}^k v_{e_i}(\sum_{j=1}^{i-1}s_{e_j})\ge  V$.

The problem is obviously at least as hard as the regular Knapsack problem (i.e., weakly NP-hard). I wonder if the problem is any harder? Can the dynamic programming solution for the regular problem be used for this problem as well.
Consider the special case where $V_i\le v_i(s)&lt; V_i+\epsilon$ such that $\epsilon$ is a very small positive constant, then your problem is equivalent to the regular knapsack problem where item $i$ has price/value $V_i$. So your problem is still NP-hard.