Let $\mathbf{x}_1, \dots, \mathbf{x}_n \in \mathbb{R}^d$ be $n$ given vectors. Define the function

$$ \mathcal{K}(\mathbf{x},\mathbf{y}) := \alpha\exp\left(-\frac{\|\mathbf{x}-\mathbf{y}\|^2}{2\sigma^2}\right) $$

where $\alpha$ and $\sigma$ are given constants. Now define the $n\times 1$ vector

$$ \mathcal{K}_n(\mathbf{x}) := \begin{bmatrix} \mathcal{K}(\mathbf{x},\mathbf{x}_1) & \dots & \mathcal{K}(\mathbf{x},\mathbf{x}_n) \end{bmatrix}^\top $$

Let $\mathbf{A}$ be any $n \times n$ positive definite matrix and $\mathbf{b}$ be any $n \times 1$ real vector. Let $\lambda > 0$ be given. Consider the optimization problem

$$ \max_{\mathbf{x} \in \mathbb{R}^d} \quad\mathcal{K}_n(\mathbf{x})^\top\mathbf{b}-\lambda\mathcal{K}_n(\mathbf{x})^\top\mathbf{A}\mathcal{K}_n(\mathbf{x}) $$

Is this optimization problem known in literature? How do I do gradient ascent on this?