I want to get the conjugate of this function when r is constant $F(r,x)=0.5\|r\|^2+\lambda\|x\|_1+\langle y,Ax-b-r \rangle$
Using the definition of the conjugate function, I know that I need to solve: $F^*(r,y)=\sup_x\{{y^T[Ax-b-r+x]+\lambda\|x\|_1}\}$
I have already found that the conjugate when x is constant is $F^*(y,x)=y^T(Ax-b)+\lambda\|x\|_1$