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.
Adly, S., Nacry, F., Thibault, L.: Preservation of prox-regularity of sets with applications to constrained optimization. SIAM J. Optim. 26(1), 448–473 (2016)MathSciNetCrossRefGoogle Scholar
Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357(4), 1275–1301 (2005)MathSciNetCrossRefGoogle Scholar
Bacák, M., Borwein, J.M., Eberhard, A., Mordukhovich, B.S.: Infimal convolutions and Lipschitzian properties of subdifferentials for prox-regular functions in Hilbert spaces. J. Convex Anal. 17(3-4), 737–763 (2010)MathSciNetzbMATHGoogle Scholar
Baillon, J.B., Bruck, R.R., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)MathSciNetzbMATHGoogle Scholar
Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596–636 (2003)MathSciNetCrossRefGoogle Scholar
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)Google Scholar
Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)MathSciNetCrossRefGoogle Scholar
Bauschke, H.H., Combettes, P.L., Noll, D.: Joint minimization with alternating Bregman proximity operators. Pac. J. Optim. 2(3), 401–424 (2006)MathSciNetzbMATHGoogle Scholar
Bauschke, H.H., Chen, J., Wang, X.: A Bregman projection method for approximating fixed points of quasi-Bregman nonexpansive mappings. Appl. Anal. 94, 75–84 (2015)MathSciNetCrossRefGoogle Scholar
Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969). (The original version appeared in Akademija Nauk SSSR. žurnal Vyčislitel’ noı̆ Matematiki i Matematičeskoı̆ Fiziki 9 (1969), 509–521.)CrossRefGoogle Scholar
Polyak, B.T.: Introduction to Optimization. Optimization Software (1987)Google Scholar
Polyak, B.T.: Random algorithms for solving convex inequalities Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 409–422. Elsevier (2001)Google Scholar
Richardson, L.F.: Measure and Integration: A Concise Introduction to Real Analysis. Wiley (2009)Google Scholar
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar