Chambolle–Pock algorithm: Difference between revisions

Revision as of 14:27, 7 July 2023

Revision as of 14:41, 7 July 2023

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==Problem statement==

Let be <math> \mathcal{X}, \mathcal{Y} </math> two real [[Vector space|vector spaces]] equipped with an [[Inner product space|inner product]] <math> \langle \cdot, \cdot \rangle </math> and a [[Norm (mathematics)|norm]] <math> \lVert \,\cdot \,\rVert = \langle \cdot, \cdot \rangle^{\frac{1}{2}} </math>. From up to now, a function <math>F</math> is called ''simple'' if its [[proximal operator]] <math> \text{prox}_{\tau F} </math> has a [[Closed-form expression|closed-form representation]] or can be accurately computed, for <math>\tau >0</math>.,<ref name=":0" /> where with <math> \text{prox}_{\tau F} </math> we refer to

:<math display="block"> x = \text{prox}_{\tau F}(\tilde{x}) = \text{arg } \min_{x'\in \mathcal{X}}\left\{

\frac{\lVert x'-\tilde{x}\rVert}{2\tau} + F(x')

\right\}</math>

Consider the following constrained primal problem:<ref name=":0" />