Clean up "equivalence" of penalty and constraint form regularization #78

There's something going on that needs clarification.
Suppose w^* is an unconstrained minimizer of phi(w), and suppose Omega(w^*)=R.
Say phi(w) is a quadratic objective for a linear system that has a null space. Then there's another minimizer w^** that has Omega(w^)=r>R.
So w^ is a solution to a constrained optimization problem, but not of the penalized form. What assumption is not met in the equivalence theorem from the homework?

Even if w^** is not attained as a solution to a penalty form problem, say something about the corresponding linear function being attained?