I Really Need Some General Advice For Laning Against Malzahar (Becoming Frustrated With Him)

Like, I don't see this Champion often enough to comment on him, but whenever I do see him, he's a serious pain in the ass to fight. The champion I'm referring to is .

First off, I would like to point out that whenever I play Mid Lane, I play , as she's really the only mid laner I know how to play decently, for that specific role. Unless the enemy team's mid laner picked before me (and they chose Zed, Katarina or Akali), then, and only then, will I pick Ahri or Annie. [basically anything that would be safer than playing Lux.]

So, for the pretty much all the times I've seen Malzahar picked, he's picked after me.

Now, with those details out of the way, I genuinely need some advice on how to deal with him. I can't shove the wave(s) against him due to his "E" auto-shoving mine back to me. Not to mention his "pets" help with this, so pushing back isn't an option. I've tried doing other things like farming jungle while my jungler is ganking other lanes. I've tried taking enemy jungle farm when I see them in other lanes. I've even tried grouping early to try and force other objectives to go down faster. None of these have been successful. Not even close.

The most I can do in lane is pop his shield with my Lucent Singularity, but in doing that, Malz just backs off and waits out my Illumination debuff, leaving me with no real way to "truly" damage him. It gets even worse when he rushes a Banshee's Veil because (for reasons I do not understand) it procs before his passive shield and not at the same time. Meaning I now HAVE to use a spell to break that before I can even consider popping his passive shield. And then, to top all of that off, if I ever DO MANAGE to pop all of those shields, he just ignite+ults me and I'm dead.

I've tried bringing against him, until I learned that you can't cleanse his ult, due to it being a Suppression. So, with all my options dead in the water, I really want to hear from anyone who's gone up against this champion and has either not been utterly beaten down by him in lane, or has actually beaten him into the ground, instead.

TL;DR - I play Lux when I play Mid Lane. Don't know how to fight Malzahar. Seeking help. Also, any advice given can be generalized between champions. Your advice doesn't necessarily have to be for the Lux vs Malzahar match-up, specifically.

Thank you all in advance!