I have found a solution algorithm for integer knapsack problems of the following form:
$\max\limits_{x_j \in [l_j,u_j]} \sum_{j=1}^n f_j(x_j)$
such that $\sum_{j=1}^n g_j(x_j) \leq b$
Where $l_j<u_j$ and $f_j$ are concave and $g_j$ are convex on $[l_j,u_j]$, and both $f_j$ and $g_j$ are monotonically increasing on $[l_j,u_j]$.
I have a problem of the following form and im trying to write it in the above form but im not sure its possible:
$\min\limits_{x_j \in [l_j,u_j]} \sum_{j=1}^n x_j c_j$
such that $\sum_{j=1}^n \frac{1}{x_j^{p-1}}d_j \leq b$
Where $p\in (1,2]$ and $c_j, d_j$ are known positive constants.
I could turn the $\min x_jc_j$ into a $\max f_j$ by defining $f_j(x_j) = -c_jx_j$ and it would be concave but the problem is the monotonicity. I cant introduce a subsitution $y_j=-x_j$ because thats not defined for the $g_j$. Is there maybe another approach?