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Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

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In search for optimal solution for a minimization problem

I am trying to solve the following optimization problem: $\begin{align}\min_{\mathbf{c}_{k}} \sum_{k=1}^{K}\frac{q_k}{c_k} \\ \text{s.t. } c_k &\geq t_k \quad \forall k \tag1\label1 \\ \sum_{k=1}^{K} …
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