maximize untility AND minimize disutility DISCLAIMER: This could very well be all bogus, I'm talking about my intuition as a beginner/intermediate economics student, you have been warned.

I got the feeling agents do not just maximize utility but minimize disutility as well. Now my understanding is that from the convexity assumption follows that those problems would have the same optimum. But I think it could very well be the case that the utility function does not at all look like the disutility function (also different variables) and optimum of those two functions need not be unique (Am I with the duality gap on the right track?).

If this at all a valid intuition I would appreciate some pointers to empirical studies that examines utility and disutiliy, or some theories that incorporate the notion of disutiliy (not just as an inverse of utility) into preferences.

P.S. I got the idea while reading about The Paradox of Choice from Barry Schwartz (more choices could lead to confusion and in my interpretation disutiliy)
I believe what you're getting at is a concept in economics called "Loss Aversion". It has been used quite extensively in recent economic research, especially in behavioral economics and finance. Another important key word here is "Prospect Theory", which encompasses loss aversion as well. 

The basic idea is that people suffer more from a loss of x\$ than they enjoy a gain of x\$ compared to a reference point. The utility function has a different (smaller in absolute value) slope when moving in a positive direction from the reference point than when moving in a negative direction. This is illustrated by the graph below (source: https://upload.wikimedia.org/wikipedia/commons/4/4e/Valuefun.jpg).



Under these conditions,  utility and disutility are different. Otherwise, maximizing untility or minimizing disutility is the same, as you mentioned. 

This insight goes back to Amos Tversky and Daniel Kahneman.Kahneman received the Nobel Prize in Economics in 2002 for this work.

A summary of the most important literature can be found for example here: http://loss-aversion.behaviouralfinance.net/