On the fuzzy sum rule for the regular subdifferential
Patrick MehlitzAug. 2, 2022, 8 a.m. S2 416-2
Motivated by the derivation of necessary optimality conditions for nonsmooth optimization problems in abstract spaces, we review available (fuzzy) sum rules for the regular subdifferential of two lower semicontinuous functions.
Most of the available literature seems to indicate that one of the involved summands needs to be locally Lipschitz continuous for the fuzzy sum rule to work. In this talk, we do away with this broadly accepted myth. For that purpose, we introduce the notion of joint lower semicontinuity of two lower semicontinuous functions and show that the latter is sufficient for applicability of the fuzzy sum rule. Some calculus reveals that joint lower semicontinuity is present whenever one of the involved summands is locally uniformly continuous which is much weaker than local Lipschitzness. Our arguments recover similar results obtained by Borwein and Zhu. By means of examples from optimal control, we visualize that joint lower semicontinuity might be present even in situations where both summands are discontinuous and use this observation to state necessary optimality conditions for optimal control problems with sparsity promoting terms in the objective function.
This talk is based on joint work with Marian Fabian (Prague, Czech Republic) and Alexander Y. Kruger (Ballarat, Australia).