proximal_convex_conj

odl.solvers.nonsmooth.proximal_operators.proximal_convex_conj(prox_factory)[source]

Calculate the proximal of the dual using Moreau decomposition.

Parameters
prox_factorycallable

A factory function that, when called with a step size, returns the proximal operator of F

Returns
prox_factoryfunction

Factory for the proximal operator to be initialized

Notes

The Moreau identity states that for any convex function F with convex conjugate F^*, the proximals satisfy

\mathrm{prox}_{\sigma F^*}(x) +\sigma \,
\mathrm{prox}_{F / \sigma}(x / \sigma) = x

where \sigma is a scalar step size. Using this, the proximal of the convex conjugate is given by

\mathrm{prox}_{\sigma F^*}(x) =
x - \sigma \, \mathrm{prox}_{F / \sigma}(x / \sigma)

Note that since (F^*)^* = F, this can be used to get the proximal of the original function from the proximal of the convex conjugate.

For reference on the Moreau identity, see [CP2011c].

References

[CP2011c] Combettes, P L, and Pesquet, J-C. Proximal splitting methods in signal processing. In: Bauschke, H H, Burachik, R S, Combettes, P L, Elser, V, Luke, D R, and Wolkowicz, H. Fixed-point algorithms for inverse problems in science and engineering, Springer, 2011.