Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 1009–1078 | Cite as

Subgradient Projectors: Extensions, Theory, and Characterizations

  • Heinz H. BauschkeEmail author
  • Caifang Wang
  • Xianfu Wang
  • Jia Xu


Subgradient projectors play an important role in optimization and for solving convex feasibility problems. For every locally Lipschitz function, we can define a subgradient projector via generalized subgradients even if the function is not convex. The paper consists of three parts. In the first part, we study basic properties of subgradient projectors and give characterizations when a subgradient projector is a cutter, a local cutter, or a quasi-nonexpansive mapping. We present global and local convergence analyses of subgradent projectors. Many examples are provided to illustrate the theory. In the second part, we investigate the relationship between the subgradient projector of a prox-regular function and the subgradient projector of its Moreau envelope. We also characterize when a mapping is the subgradient projector of a convex function. In the third part, we focus on linearity properties of subgradient projectors. We show that, under appropriate conditions, a linear operator is a subgradient projector of a convex function if and only if it is a convex combination of the identity operator and a projection operator onto a subspace. In general, neither a convex combination nor a composition of subgradient projectors of convex functions is a subgradient projector of a convex function.


Approximately convex function Averaged mapping Cutter Essentially strictly differentiable function Fixed point Limiting subgradient Local cutter Local quasi-firmly nonexpansive mapping Local quasi-nonexpansive mapping Local Lipschitz function Linear cutter Linear firmly nonexpansive mapping Linear subgradient projection operator Moreau envelope Projection Prox-bounded Proximal mapping Prox-regular function Quasi-firmly nonexpansive mapping Quasi-nonexpansive mapping (C,ε)-firmly nonexpansive mapping Subdifferentiable function Subgradient projection operator 

Mathematics Subject Classification (2010)

Primary 49J52 Secondary 49J53 47H04 47H05 47H09 


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The authors thank two anonymous referees for careful reading and constructive suggestions on the paper. HHB was partially supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Canada Research Chair Program. CW was partially supported by National Natural Science Foundation of China (11401372). XW was partially supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada (NSERC). JX was supported by by NSERC grants of HHB and XW.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
    Email author
  • Caifang Wang
    • 2
  • Xianfu Wang
    • 1
  • Jia Xu
    • 1
  1. 1.MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.Department of MathematicsShanghai Maritime UniversityShanghaiChina

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