= Norm convergence of elastic net regularization
Assume an exact solution of finite penalty exists and select the unique $J$-minimizing exact solution $u^\dagger$. Minimality gives $\|Ku_\alpha^\delta-f^\delta\|^2/2+\alpha J(u_\alpha^\delta)\leq\delta^2/2+\alpha J(u^\dagger)$. If $\alpha\to0$ and $\delta^2/\alpha\to0$, boundedness, <weak sequential compactness of bounded sequences in a reflexive Banach space> and lower semicontinuity identify every weak limit with $u^\dagger$. The separate lower semicontinuity of $\|u\|_1$ and $\|u\|_2^2$, together with the upper bound on $J$, forces the Hilbert norms to converge. The <Radon-Riesz theorem> then gives <norm convergence>.
Back to article page
