Abstract
Motivated by the recent works on proximity operators and isotone projection cones, in this paper, we discuss the isotonicity of the proximity operator in quasi-lattices, endowed with general cones. First, we show that Hilbert spaces, endowed with general cones, are quasi-lattices, in which the isotonicity of the proximity operator with respect to one order and two mutually dual orders is then, respectively, studied. Various sufficient conditions and examples are introduced. Moreover, we compare the proximity operator with the identity operator with respect to the orders. As applications, we study the solvability and approximation results for the nonconvex nonsmooth optimization problem by the order approaches. By establishing the increasing sequences, we, respectively, discuss the region of the solutions and the convergence rate, which vary with combinations of the mappings, and hence, one can choose the proper combination of the mappings under specific conditions. Compared to other approaches, the optimal solutions are obtained and inequality conditions hold only for comparable elements with respect to the orders.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Moreau, J.J.: Les liaisons unilatérales et le principe de Gauss. C. R. Acad. Sci. Paris A256, 871–874 (1963)
Moreau, J.J.: Sur la naissance de la cavitation dans une conduite. C. R. Acad. Sci. Paris A259, 3948–3950 (1964)
Moreau, J.J.: Quadratic programming in mechanics: dynamics of one-sided constraints. SIAM J. Control 4, 153–158 (1966)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)
Byrne, C.L.: Iterative Optimization in Inverse Problems. CRC Press, Boca Raton (2014)
Combettes, P.L., Müller, C.L.: Perspective functions: proximal calculus and applications in high dimensional statistics. J. Math. Anal. Appl. 457, 1283–1306 (2018)
Combettes, P.L., Salzo, S., Villa, S.: Consistent learning by composite proximal thresholding. Math. Program. B167, 99–127 (2018)
Sra, S., Nowozin, S., Wright, S.J. (eds.): Optimization for Machine Learning. MIT Press, Cambridge (2012)
Vaiter, S., Peyré, G., Fadili, J.: Model consistency of partly smooth regularizers. IEEE Trans. Inf. Theory 64, 1725–1737 (2018)
Combettes, P.L., Glaudin, L.E.: Proximal activation of smooth functions in splitting algorithms for convex image recovery. arXiv:1803.02919v3 (2018)
Isac, G., Németh, A.B.: Monotonicity of metric projections onto positive cones of ordered Euclidean spaces. Arch. Math. 46(6), 568–576 (1986)
Németh, S.Z., Xiao, L.: Linear complementarity problems on extended second order cones. J. Optim. Theory Appl. 176, 269–288 (2018)
Ferreira, O.P., Németh, S.Z.: On the spherical convexity of quadratic functions. J. Glob. Optim. 73, 537–545 (2019)
Ferreira, O.P., Németh, S.Z., Xiao, L.: On the spherical quasi-convexity of quadratic functions. Linear Algebra Appl. 562, 205–222 (2019)
Abbas, M., Németh, S.Z.: Finding solutions of implicit complementarity problems by isotonicity of the metric projection. Nolinear Anal. 75(4), 2349–2361 (2012)
Li, J.: Isotone cones in Banach spaces and applications to best approximations of operators without continuity conditions. Optimization 67(5), 563–583 (2018)
Kong, D., Liu, L., Wu, Y.: Characterization of the cone and applications in Banach spaces. Numer. Func. Anal. Optim. 40, 1703–1719 (2019)
Németh, S.Z., Zhang, G.: Extended Lorentz cone and variational inequalities on cylinders. J. Optim. Theory Appl. 168, 756–768 (2016)
Ferreira, O.P., Németh, S.Z.: How to project onto extended second order cones. J. Glob. Optim. 70, 707–718 (2018)
Kong, D., Liu, L., Wu, Y.: Isotonicity of the metric projection by Lorentz cone and variational inequalities. J. Optim. Theory Appl. 173, 117–130 (2017)
Bello Cruz, J.Y., Ferreira, O.P., Németh, S.Z., Prudente, L.F.: A semi-smooth Newton method for projection equations and linear complementarity problems with respect to the second order cone. Linear Algebra Appl. 513, 160–181 (2017)
Messerschmidt, M.: Normality of spaces of operators and quasi-lattices. Positivity 19, 695–724 (2015)
Li, J., Ok, E.: Optimal solutions to variational inequalities on Banach lattices. J. Math. Anal. Appl. 388, 1157–1165 (2012)
Zhang, Z.T.: Variational, Topological, and Partial Order Methods with Their Applications. Springer, Berlin (2012)
Guo, D., Cho, Y., Zhu, J.: Partial Ordering Methods in Nonlinear problems. Nova Science Publishers Inc, New York (2004)
Acknowledgements
The authors would like to thank the referee for his/her very important comments that improve the results and the quality of the paper. The authors were supported financially by the National Natural Science Foundation of China (11871302, 71773067), the Australian Research Council, the Natural Science Foundation of Shandong Province of China (ZR2017MA034) and Key R&D Program of Shandong Province (2019GGX101024).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Sándor Zoltán Németh.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kong, D., Liu, L. & Wu, Y. Isotonicity of Proximity Operators in General Quasi-Lattices and Optimization Problems. J Optim Theory Appl 187, 88–104 (2020). https://doi.org/10.1007/s10957-020-01746-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-020-01746-2