There's another question that answers this though it doesn't specify anything regarding proxy authentication.
Its solution is
(setq url-proxy-services '(("no_proxy" . "work\\.com")
("http" . "proxy.work.com:911")))
There's another question that answers this though it doesn't specify anything regarding proxy authentication.
Its solution is
(setq url-proxy-services '(("no_proxy" . "work\\.com")
("http" . "proxy.work.com:911")))
ELPA uses the "url" package. As far as I know there is no way to do proxy authentication with it.
Can you set up your proxy auth outside of emacs?
Well, if you really want to do this and do not mind using another program then ... socat is the answer. Use socat to forward a local port through to a connection passing through the http proxy. You are not bypassing it, just "bolting on" the functionality to an application that does not have it (in case anyone asks). This might be difficult.
Another solution that would work great if you are on a unixy OS is to install your own non-authenticating http proxy that uses the authenticating proxy (like squid). This might look like circumvention to some people. Be careful.
For example, take a look at Proxytunnel.
UPDATE: Mike Hoss seems to be correct in the comment he adds to the question linked to above. The URL package will ask for id and password. At least that is what I see in the defun for url-http-create-request in file url-http.el.