First of all let me say that I'm new to optimization. I realized that quasi-convex functions share with convex functions some nice properties, so I wonder if we can push the weakening a little further. We all know that quasi-convex functions are functions whose sublevel sets are convex. What if we try to build an optimization theory for objective functions whose sublevel sets are...
- bounded
- simply connected
- star-shaped
- combinations of above... or anything else.
I'm curious if is there a direction that lead to satisfactory results. Ok, nothing can compare to convex optimization, but some results for more general functions may be interesting. Thanks in advance!
Minimizing continuous functions over compact level-sets is a nice problem.
Boundedness of level-sets alone is not enough. The function $$ f(x) = \begin{cases} 1-x^2 & |x|<1\\ x^2 & |x|\ge 1 \end{cases} $$ has bounded level sets, but there exists no global minimizer.
The function $f(x)=e^x$ has star-shaped, connected level-sets, but no minimizer exists.