The current Lightning Network specification allows channel peers to specify a base fee and a proportional fee. In other words, the fee charged is a linear function of the forwarded amount. While that sounds completely reasonable, my question is, have there been any other proposed fee functions, perhaps to more closely align with node operator incentives, or to make path calculations easier? (Feel free to include the "zero base fee" proposal for completeness.)
I'm not familiar with any proposals to add new variables to the fee function. There have been discussions on HODL invoices which would (if enabled on the Lightning Network) require additional fees proportional to the time the capital was going to be locked up for. I'm not sure what the latest status of HODL invoices is, their introduction would compete with (and potentially negatively impact) the current use case of instantaneous payments but some form of them seems to have been merged into LND.
Most of the discussion I am aware of has been about removing the base fee and just having the proportional fee as this would better facilitate some proposed path finding algorithms.