For this optimization problem:
$$\min_w w^T\delta, \text{such that }\sum_iw_i = q, w_i \ge 0, \delta_i \ge0$$
My intuition says that the minimization of this problem is simply assigning the constrained value $q$ to the minimum $\delta_i$. Formally, I want to prove the following inequality:
$$q\delta_1 \le w_1\delta_1 + w_2\delta_2 \le w_1\delta_1 + w_2\delta_2 + w_3\delta_3 \le \dots \le w_1\delta_1 + w_2\delta_2 + \dots + w_n\delta_n$$
where $\delta_i$'s are ordered in ascending order (i.e., $\delta_1$ is the smallest), and the summation of weights for each term is $q$, so $w_1 + w_2 = q$ in the second term of the inequality.
Is my intuition correct? If so, any leads where to start?
Good intuition.
We have
\begin{align} \sum_{i=2}^n w_i\delta_i & \ge \sum_{i=2}^n w_i\delta_1 \\ &= \delta _1 \left(\sum_{i=2}^n w_i\right) \\ &= \delta_1 (q-w_1) \end{align}
Rearrange the inequality and you should be able to reach your conclusion.
Interpretation:
View $\delta_i$ as the unit product price by the $i$-th supplier and we want to purchase $q$ units of products, to minimize the cost, purchase all at the cheapest price.