[chang_ramsey] Fix Runtime and Deprecation warnings

[kp992]

Fix Runtime and Deprecation warnings

• Avoid invalid inputs by checking and clipping the limits.
• Fix API deprecation warnings by using new API formats.

Fixes #201

[mmcky]

nice work! Thanks @kp992 -- I'll review this afternoon and merge.

[mmcky]

@kp992 should we install cvxopt in this environment?

[mmcky]

@kp992 it looks like we have introduced some new Runtime Warnings. Would you mind to take a look?

[kp992]

Thanks @mmcky. The first one is not really a warning but that could be fix by installing cvxopt. I was reading more about the second one and looks like its coming from the internals of scipy and it's something I think not in our hand.

[mmcky]

thanks @kp992 -- here is what Gemini had to say about it :-)

The "Values in x were outside bounds during a minimize step, clipping to bounds" warning in scipy.optimize arises when the optimization algorithm attempts to evaluate the objective function at a point outside the defined bounds. To resolve this, consider the following:

Initial Guess:
Ensure the initial guess x0 lies within the specified bounds. If not, the algorithm might struggle to find a feasible solution.

Bounds Definition:
Verify the bounds are correctly defined, especially for multiple variables. Use (min, max) tuples for each variable.

Algorithm Choice:
Some algorithms handle bounds more effectively than others. If using an algorithm like L-BFGS-B, which explicitly supports bounds, the issue might stem from other factors. Consider trying different algorithms.

Objective Function:
Check if the objective function itself might be causing the issue. For instance, numerical instability or undefined behavior outside the bounds could trigger the warning.

SciPy Version:
In some cases, this warning has been linked to specific SciPy versions. Upgrading or downgrading SciPy could potentially resolve the issue.

If the warning persists despite these checks, it might be benign and not affect the final result, especially if the clipping occurs only during intermediate steps. However, it's crucial to ensure the solution remains within the bounds and that the warning doesn't indicate a more serious problem with the optimization process.

Not bad advise :-)

[kp992]

Thanks @mmcky, I already tried different algo like L-BFGS-B but they don't support constraints which will lead to new set of warnings.