FAQ: norms in optimization problems behave weird

In textbooks and papers, it is very common to encounter norms (L1, L2, Linf) in the objective of optimization problems.

While casadi implements operations for norms (Python: norm_1(.), norm_2(.), norm_inf(.), Matlab: norm(.,1),norm(.,2),norm(.,inf)`), you most likely do not want to use these directly in an optimization problem, since they are nonsmooth.

Instead, reformulate your problem. Instead of optimizing for || . ||_2, optimize for || . ||_2^2 (e.g. using sumqr(.)). Instead of optimizing with || . ||_1 or || . ||_inf, do a slack variable reformulation into a smooth problem.

Related, the derivative of the L2 norm is a fragile concept: when the argument reaches zero, the derivative is of the form 0/0 resulting in a NaN. Consider the 1-D case: L2 norm of x becomes sqrt(x), its derivative is 1/(2*sqrt(x)) which is problematic for x=0. See also https://github.com/casadi/casadi/wiki/FAQ:-Why-am-I-getting-%22NaN-detected%22in-my-optimization%3F