After a short survey of the main results
and open questions on norm attaining
operators (Lindenstrauss properties "A"
and "B", renormings, connection with the
Radon-Nikodym property) we will consider
some related optimization problems. More
concretely, we will discuss the denseness
of norm attaining multilinear forms or
polynomials on a Banach space, with special
attention to classical Banach spaces.