Partial regularization of the sum of two maximal monotone operators

A greedy approach to sparse poisson denoising

“…The last term, called bias, quantifies the error we make by using the Moreau-Yosida regularization instead of the Poisson neg-loglikelihood. Assuming we could bound beforehand the maximum intensities of the x i , x o and x λ , we could use a result in [14] showing that α λ − α o = O( √ λ) which quantifies this error (remember that the Moreau-Yosida regularization converges to the original function when λ goes to 0).…”

On the convergence of the forward-backward algorithm for null-point problems

“…where C λ is the Yosida approximate of C. This is a relevant approximation, since in addition to the fact that C λ is λ -cocoercive, we have that B+C λ graph converges to B+C when B+C is a maximal monotone operator and it was established, for example, that if x * is a limit point of the family {x * λ , λ → 0} and if we assume that {C λ (x * λ ), λ → 0} is bounded, then x * solve (3.3), see [5]. If in addition C is strongly monotone and x * the soultion of (3.3), by [5]-Theorem 3, we also have the following estimate…”

Systems of Structured Monotone Inclusions: Duality, Algorithms, and Applications

“…In turn, p 0 = − K k=1 p k = 0, i.e., x ∈ A −1 0. Altogether, x ∈ Z. which itself covers the frameworks of [10,18,35,37] and the references therein. In this case, Proposition 4.2 specializes to [17,Proposition 6.10].…”