ATTOUCH--THERA DUALITY, GENERALIZED CYCLES, AND GAP VECTORS

Using the Attouch--The'\ra duality, we study the cycles, gap vectors, and fixed point sets of compositions of proximal mappings. Sufficient conditions are given for the existence of cycles and gap vectors. A primal-dual framework provides an exact relationship between the cycles and gap vectors. We also introduce the generalized cycle and gap vectors to tackle the case when the classical ones do not exist. Examples are given to illustrate our results.

Automated summary

Using the Attouch-The'\ra duality, cycles, gap vectors, and fixed point sets of compositions of proximal mappings are studied, with sufficient conditions given for the existence of cycles and gap vectors. A primal-dual framework establishes an exact relationship between the cycles and gap vectors, and generalized cycle and gap vectors are introduced for cases where classical ones do not exist. Examples are provided to illustrate the results.