Stop any service and create a "snapshot". Since my Proxmox doesn't have a snapshot solution yet I specifically choose to use the local (SSD) storage without compression to get the backup to finish as fast as possible, we can't afford unnecessary downtime and in the next generation of our servers we will be using file system based snasphots (LVM or ZFS) which makes waiting for a backup to finish something of the past. I created my snapshot using the web interface of Proxmox 3.4 but this is what happens in the background after stopping the container.

vzdump 113 --remove 0 --mode stop --storage local --node duff-prox-01

I changed the IP of my VPS during the entire upgrade.

sed -i "s/old_ip/temporary_ip/g" /etc/network/interfaces /etc/hosts
/etc/init.d/networking restart

Next upgrade Ubuntu (this application is in the update-manager-core package)

do-release-upgrade

You will get a warning and we will address this by replacing Systemd with the legacy Upstart before rebooting in Ubuntu 16.04

apt-get install upstart-sysv

After the reboot, check with netstat if everything started fine, change the IP again and reboot to start using a fully up to data Ubuntu in a dated VPS environment.

netstat -atnp 
apt-get -y upgrade
apt-get -y dist-upgrade
sed -i "s/temporary_ip/old_ip/g" /etc/network/interfaces /etc/hosts
reboot