apt-get update warning: Ignoring Provides line with DepCompareOp for package…

This problem happens because your version of apt is "too old" and doesn't support packages that use versioned "Provides:".

You can check your apt version with

apt --version

"versioned provides" were added in apt 1.0.7, but Ubuntu 14.04 ships with apt 1.0.1.

See https://bugs.launchpad.net/ubuntu/+source/libreoffice/+bug/1662227 for more information.

The warning is harmless. If you really want to remove it, you can compile apt from sources to a version that supports versioned provides.

Warning!: Updating apt is potentially dangerous and can break the package manager.

If you choose to do it, follow these steps:

wget http://snapshot.debian.org/archive/debian/20150821T213547Z/pool/main/a/apt/apt_1.0.10.2.tar.xz
tar -xvf apt_1.0.10.2.tar.xz
cd apt-1.0.10.2
./configure
make

The real solution appears to track down the offending package and ask the PPA maintainer to fix it.

Since the issue happened because of certbot, I opened issue 6063 to tackle this. This comment asked the maintainer of the PPA to fix the problem.

As off Monday 23 October, this has been resolved. Make sure you do

sudo apt-get update
sudo apt-get upgrade

To get the changes.