so I want to find a non-trivial lower bound on the following integer optimization problem. We have a vector $\mathbb{\alpha} = (\alpha_1,...,\alpha_C) \in \mathbb{R}_+^C$ that fulfills the constraint $\sum_{i=1}^{C} \alpha_i = 1$ and another vector $\mathbb{\beta} = (\beta,...,\beta) \in \mathbb{R}_+^K$ that also sums up to 1. The optimization variable is a boolean matrix $Y\in \{0,1\}^{C \times K}$. We can assume that $K > C$, typically $K$ will be in range $[20,50]$ and $C$ in $ [10, 15]$.
The problem is the following: $$ \text{minimize:} \ \sum_{i=1}^{C} \Big[ \big( \sum_{k=1}^K \beta_k \frac{\alpha_i Y_{i,k}}{\sum_{l=1}^C \alpha_l Y_{l,k}} \big) - \alpha_i \Big]^2$$
$$\text{subject to:} \ \sum_{k=1}^K Y_{i,k} \geq 1 \quad \forall i \in \{1,...,C \} $$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{i=1}^C Y_{i,k} \geq 1 \quad \forall k \in \{1,...,K \}$$
Now I can obviously not try out the exponentially growing number of solutions, so I was wondering if there is a standard way to get lower bounds on such problems?
I already tried a simple continuous relaxation which computes a matrix $F \in [0,1]^{C \times K}$ where $F_{i,k} \approx \frac{\alpha_i Y_{i,k}}{\sum_{l=1}^C \alpha_l Y_{l,k}}$, but with all the constraints I could derive there always exists a solution with objective value 0, so I need to find another way to do this. I am however not very familiar with this type of discrete optimization so I am looking for some methods that could turn out to be valuable.
Thank's a lot.
Unless I am making an error, setting $Y_{i,k}=1\,\forall i,\forall k$ satisfies the constraints and produces an objective value of 0, which must be optimal.