What is the minimum of this weighted sum, considering all constraints?

Question

I have started to self-study some non-linear programming courses. In one of the references I have encountered a challenging problem which I was unable to solve it: Given a number $q, 0 < < 1$, and a set of positive weights ${_1, _2, … . _}$, the goal is to minimize the weighted sum $S=\varSigma_{k=0}^n\,\alpha_kq^{x_k}$.The the sum of all $x_k$ must be less than a total budget T. Moreover, all $x_k$ are non-negative. Also, $x_k$ are non-negative integers that each satisfy some local budget $_ (i.e., 0 ≤ x_k ≤ )$. Find the optimal value for $x_k$.

Answer 1

One way to solve this is to linearize the objective and use an integer linear programming solver. For $k \in \{1,\dots,n\}$ and $i \in \{0,\dots,T_k\}$, introduce binary variable $y_{k,i}$ to indicate whether $x_k = i$. The problem is to minimize $$S=\sum_{k=0}^n \alpha_k \sum_{i=0}^{T_k} q^i y_{k,i}$$ subject to linear constraints: \begin{align} \sum_{i=0}^{T_k} y_{k,i} &= 1 &&\text{for all $k$}\\ \sum_{i=0}^{T_k} i y_{k,i} &= x_k &&\text{for all $k$}\\ \sum_{k=0}^n x_k &\le T \end{align}