Maximizing a sum of Gaussians 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?
 A: I don't see any connection between your problem and MPC.
Take a look at optimization via the technique known as stochastic approximation; it is extremely popular today for several reasons. Check out "Optimization Methods for Large-Scale Machine Learning" by Bottou, Curtis, and Nocedal on arXiv: https://arxiv.org/abs/1606.04838, especially Section 3.
A: You are looking for a stochastic optimizer [1].
This optimization is stochastic. Thus, it requires you optimize the expectation $E[.]$ of a particular value.
In an engineering problem, they often have some constraint on top of optimization which form a stochastic programming problem [2][3][4]. In such problems, the constraints also become stochastic.
I refer you to a great publication on Stochastic Model Predictive Control:


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*Mesbah, A., 2016. Stochastic model predictive control: An overview and perspectives for future research. IEEE Control Systems, 36(6), pp.30-44. [5]
To make the life easy, you can use a Monte Carlo method with a bunch of scenarios to perform an optimization. I refere you to a few publications:


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*Janson, L., Schmerling, E. and Pavone, M., 2018. Monte Carlo motion planning for robot trajectory optimization under uncertainty. In Robotics Research (pp. 343-361). Springer, Cham. [6]

*Maldonado, D.A., 2017. Sequential Monte Carlo Methods for Parameter Estimation, Dynamic State Estimation and Control in Power Systems (Doctoral dissertation). [7]

*Oldewurtel, F., Jones, C.N., Parisio, A. and Morari, M., 2014. Stochastic model predictive control for building climate control. IEEE Transactions on Control Systems Technology, 22(3), pp.1198-1205. [8]
