There are two ways at looking at your question, already in the answers you'll see what I mean. I think better terminology would have helped as some answers seem to be going at it from the final bevel point of view and others from the main grind point of view. I'm pretty sure you mean the latter as getting a concave final bevel would be an interesting and probably fruitless attempt. The confusion likely stems from the "multi-bevel" part of your question as sharpening gurus will tout the benefits in certain instances of micro bevels. Please excuse the very basic drawing, it'll just help with visualizing what I'm about to waffle on about!
This shows a micro bevel at the edge, also technically a mutli-bevel, but I'll be referring to the main grind.
As was mentioned, it all depends on what you're cutting and how. For example:
Large kitchen knives for vegetable prep, an understated, full height convex is your friend.
General day to day utility in a folder/small fixed blade - honestly a Moo point, you'll likely never notice the difference but I'd lean towards hollow and full flat.
Choppers (I mean competition grade wood-whacker types) convex again, basically a crossbreed between knife and axe for obvious reasons.
As far as multi-bevel grinds like Loveless' duplex grind, it all depends. If it makes sense to you instantaneously and you can see why it was done, it'll probably suit you. Some makers, do multi-bevels just for the aesthetic value. Multiple bevels are likely to add drag when slicing and/or force you to only use a short section of the blade at any one time. To get the faceted look, their thicknesses need to vary and on a couple knives I've seen it cause the grind with the lowest height to be very, very chunky and for all intents and purposes useless.
I'll give a few broad pros and cons but you need to realize there are so many determining factors in each category eg:
Hollow grinds
A hollow grind achieved with a small diameter wheel will perform very differently from a hollow grind from a large diameter wheel.
Same thing but in the reverse for convex grinds (closer to flat will slice better than closer to an axe)
Multi bevels - are they all the same type? one hollow and one flat/convex or any imaginable combination?
All of the grinds will be affected by their height. The higher the grind height, the "slicier" it will be. Simple physics here, thinner profiles will part material easier than thicker profiles.
So for a couple broad strokes:
Hollow grinds
One of the easiest grinds to achieve, thus why factory knives often sport them

When cutting, the first bit of the blade entering the material is one of the thinnest so for shallow cuts they are fantastic. As the blade moves deeper, the accelerating curves of the grind cause binding. For a real world example, go cut a potato with a classic hollow grind, easy at first but requiring more effort the deeper you cut.
As I said, on a daily basis the average Joe won't notice this, all depends what you're cutting...
Grind height is a factor, Dozier grinds were brought up for good reason

Sharpening is pretty easy as on a properly ground blade there is quite a bit of thin area that will be very easy to touch up.
Flat grinds
I like flat grinds, they part material really well, they're both thin behind the edge and retain a lot of strength as less material is removed when compared to hollow grinds. They are harder to achieve as when grinding by hand, even a slight shift of pressure can completely ruin it. Machines can give you consistant results but I make my stuff by hand so my opinions are based on my experience.
Sharpening's pretty easy but the additional steel means it likely thickens faster than a hollow.
Convex grinds
Also very hard to do and very much dictated by the intended end use. I like paper thin convex grinds on my gyutos but you need a lot more meat on a chopper. if not done right they wedge like hell, sharpening is a point of contention as some feel that stropping will remove material over the whole surface maintaining a sharp edge but to my mind, their "life" is short as after a few good sharpens you're going to be in some pretty thick steel. You could have it re-ground by a maker or even attempt it yourself but at some point it will likely need a touch up.
Multi bevels
Too big a can of worms for me to even attempt

Far too many factors! In general, probably more gimmick than real-world but there are some out there that make sense. Broadly speaking, you can apply the same pros and cons I've listed depending on what grinds are used but you have to add in the probable friction caused in the transition zones.
Hope this helps!
Gazza