Approximating the simulation time

I use cvx to optimize the battery hourly schedule of one year. CVX can get the solution of optimizing about 8000 thousand variables in about 3 minutes using MOSEK solver which is very fast.

Is there any way to tell how much time cvx needs to finish the iterations?

For example, if I run the simulation for optimizing over 10 years can I ask the MATLAB to print how much time is left to get the optimal solution or how close the simulation is to finish? I am designing Graphical user interfaces and I want to tell the user how much time is left to get the optimal solution.

Experiment with different numbers of variables and see whether there is a reliable pattern.

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Thank you so much @Mark_L_Stone .
At least can I print the best objective function that has been achieved for every iteration?
I see MATLAB print the POBJ for every iteration which represents the best objective function. But I have NO idea how to access this value while running. Any idea, please?

You may have to switch to use Mosek directly if you want to access low level stuff of the optimizer.

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I see but I am designing an app using MATLAB.

Thanks a lot.

It is possible to call MOSEK directly from Matlab since that is what cvx is doing. See