Difference between Half Quadratic vs Quadratic

Question

Half quadratic minimization/penalty/optimization, I am unable to find any related material/resources.

If anyone can point to some useful resources, it will be great

Answer 1

Assume you have the following minimization problem

$ F(u)=\|Au-v\|^2_2 +\lambda \rho(\| Du\|)$

with $\rho$ a convex potential function (e.g. $\rho(t)=\sqrt{t^2+\alpha}$ , $\alpha>0 $ is a parameter) and $A$ a linear operator. Minimizing the problem with a Newton like method involves the computation of $\nabla ^2 F(u)$ which is numerically unstable if $A$ is not well conditioned. Then the main idea of half-quadratic regularization is to introduce an auxiliary variable $b$ and construct an new functional $\mathcal F(u,b)$ that is quadratic in $u$ and seperable in $b$ such that

$(\hat u, \hat b)= \arg \min \mathcal F(u,b) \rightarrow \hat u = \arg \min F(u)$

As a reference I recommend

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