DOC: Update derivation equations of positive Group Lasso penalty

[Aishwarya0811]

Context of the PR

Contributions of the PR

• Complete mathematical derivation for the subdifferential case w = 0 including full Lagrangian formulation
• Rigorous KKT conditions and complementary slackness analysis with step-by-step mathematical proof
• Replacement of incomplete placeholder text with actual mathematical derivations from the reference document
• Improved mathematical notation consistency throughout the tutorial
• Complete mathematical derivation for the subdifferential case w = 0 including full Lagrangian formulation
• Rigorous KKT conditions and complementary slackness analysis with step-by-step mathematical proof
• Replacement of incomplete placeholder text with actual mathematical derivations from the reference document
• Improved mathematical notation consistency throughout the tutorial

Checks before merging PR

• added documentation for any new feature
• added unit tests
• edited the what's new (if applicable)

[Badr-MOUFAD]

Thanks @Aishwarya0811 for the PR!

You mistakenly pushed the files _double_quadratic.py and test_double_quadratic.py, ... which are unrelated to this PR, can you remove them?

[Aishwarya0811]

Hi @Badr-MOUFAD!
Got it fixed! I removed those unrelated files - they were from my previous PR and somehow got mixed in here.
The PR is now clean and only has the documentation updates for issue #243.
Tests are all green, so it should be good to go. Let me know if you need any tweaks!

Thanks for the quick feedback 👍

[malakazlan]

documentation with formula is nicely done and correctly now .Thanks for contribution.

[Aishwarya0811]

Thank you @malakazlan for review on PR 😀

[Aishwarya0811]

@mathurinm @Badr-MOUFAD
I’ve cleaned up the PR as requested — it now only includes the updated derivations for positive Group Lasso (issue #243).
Would you mind taking another look and approving when you get a chance?
Thanks! 🙏