I posted a following question in MSE, but I think it should be posted here in MO. Since I don't know how to transfer the post from MSE to MO, I have pasted the question below. Thank you in advance and looking for your comments/suggestions.

My question is two fold:

(1) Is it possible to "solve" (iterative convex/non-convex) optimization problems via learning (e.g., regression / classification) with an objective to "solve" (in some sense) the problem in "one-shot"?

(2) If the above answer is affirmative, do you have any preferred methods or papers that you can suggest or refer to?

ADD:

Let's consider a following convex optimization whose solution is obtained iteratively via some solver (e.g., CVX).

\begin{aligned} & \underset{\mathbf{x} \in \mathbb{R}^n}{\text{minimize}} & & \left\| \mathbf{y} - \mathbf{x} \right\|_2^2 %\\ & \text{subject to} & & \mathbf{A} \mathbf{x} \leq \mathbf{b} \ ,\\ %&&& X \succeq 0. \end{aligned} where the inequality constraint is element-wise. The matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, vector $\mathbf{b} \geq 0\in \mathbb{R}_{+}^{m}$, and the vector $\mathbf{y} \in \mathbb{R}^{n}$ are given/known. Also, $n > m$, and $n$ can be a very large value.

Question is: can we utilize some "learning" to "predict" the optimal solution (non-iteratively)?

I found some papers, e.g., Link, that attempt to solve the optimization problems utilizing "learning" (e.g., neural networks). Does anyone have any feeling about this? I will try to dig into this more in the future, but would be happy to hear your experience if any.