Let $\alpha\in [0,1]^K: \sum_{i=1}^K\alpha_i=1$, and assume we have $K$ vectors $\{x(1),...,x(K)\}$ such that $x(i)\in\mathbb{R}^d, \forall i \in [K]$.
Consider the Gram matrix $G(\alpha) = \sum_{i=1}^K \alpha_i x_i x_i^T$.
Now consider the optimisation problem for $y\in\mathbb{R}^d$
$$\sup_{\alpha\in[0,1]^K: \sum_{i=1}^K\alpha_i=1} y^\top G(\alpha)y. $$
Does this problem admit a unique solution, many solutions, no solutions?
I think it should have a unique solution since we have the maximum of $G(\alpha)$ that is a polynomial function of $\alpha$ over a compact set, but I am not sure about it.
How is it possible to solve it?
Let $\Delta \subset \Bbb R^K$ denote the simplex of dimension $K-1$. This problem can be more simply rewritten as $$ \sup_{\alpha \in \Delta} \sum_{i=1}^k \alpha_i(y^Tx_i)^2. $$ It is easy to see that this maximum is attained with $\alpha_j = 1$ and $\alpha_i = 0$ for all $i \neq j$, where $j = \operatorname{argmax}_i (y^Tx_i)^2$. This is the unique point at which the maximum is obtained if and only if $j$ is the unique maximizing index.