Let V be a finite set with |V|=n. A family F Í 2^V\mathcal{F}\subseteq 2^{V} is called laminar if for arbitrary two sets X, Y Î FX, Y \in \mathcal{F}, X ∩ Y ≠ ∅ implies X ⊆ Y or X ⊇ Y. Given a laminar family F\mathcal{F}, a demand function d →ℝ₊, and a monotone concave cost function F : \mathbbR₊^V ® \mathbbR₊F : \mathbb{R}_{+}^{V} \rightarrow \mathbb{R}_{+}, we consider the problem of finding a minimum-cost x Î \mathbbR₊^Vx \in \mathbb{R}_{+}^{V} such that x(X)≥ d(X) for all X Î FX \in \mathcal{F}. Here we do not assume that the cost function F is differentiable or even continuous. We show that the problem can be solved in O(n ² q) time if F can be decomposed into monotone concave functions by the partition of V that is induced by the laminar family F\mathcal{F}, where q is the time required for the computation of F(x) for any x Î \mathbbR₊^Vx \in \mathbb{R}_{+}^{V}. We also prove that if F is given by an oracle, then it takes W(n²q){\it \Omega}(n^{2}q) time to solve the problem, which implies that our O(n ² q) time algorithm is optimal in this case. Furthermore, we propose an O(n log² n) algorithm if F is the sum of linear cost functions with fixed setup costs. These also make improvements in complexity results for source location and edge-connectivity augmentation problems in undirected networks. Finally, we show that in general our problem requires W(2^n/₂q){\it \Omega}(2^{n \over 2}q) time when F is given implicitly by an oracle, and that it is NP-hard if F is given explicitly.