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?

  • $\begingroup$ yes, still interested in it. I had given up any attempts on it though :). $\endgroup$ Commented May 23, 2022 at 20:55

2 Answers 2


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.


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:

  • 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:

  • 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]
  • 2
    $\begingroup$ Why am I looking at stochastic optimization. There is nothing random here. Everything is deterministic. $\endgroup$ Commented Aug 4, 2018 at 15:56
  • $\begingroup$ @dineshdileep. I see. How do you know this optimization is positive definite? Is it is not positive definite, a gradient descent does not necessarily work. Then, you may go after evolutionary-based methods such as Genetic Algorithm (GA). $\endgroup$
    – Arash
    Commented Aug 5, 2018 at 4:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.