Douglas–Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems

Mathematical Programming

Volumn 159, Issue 1-2, 2016, Pages 371-401

  Li, Guoyin ^a   Pong, Ting Kei ^b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

Author keywords

90C06; 90C26; 90C90

Indexed keywords

ALGEBRA; LINEAR SYSTEMS; NUMERICAL METHODS; OPTIMIZATION; SET THEORY;

ALTERNATING PROJECTION METHOD; CONVERGENCE PROPERTIES; FEASIBILITY PROBLEM; LIPSCHITZ CONTINUOUS; NONCONVEX OPTIMIZATION; NONCONVEX OPTIMIZATION PROBLEM; NUMBER OF ITERATIONS; OPTIMIZATION PROBLEMS;

PROBLEM SOLVING;

EID: 84948169278     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-015-0963-5     Document Type: Article

References (32)

1. Aragón, A.F., Borwein, J.M.: Global convergence of a non-convex Douglas–Rachford iteration. J. Global Optim. 57, 1–17 (2012)
2. Aragón, A.F., Borwein, J.M., Tam, M.K.: Douglas–Rachford feasibility methods for matrix completion problems. ANZIAM J. 55, 299–326 (2014)
3. Aragón, A.F., Borwein, J.M., Tam, M.K.: Recent results on Douglas–Rachford methods for combinatorial optimization problems. J. Optim. Theory Appl. 163, 1–30 (2014)
4. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka–Lojasiewicz inequality. Math. Oper. Res. 35, 438–457 (2010)
5. Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods. Math. Program. 137, 91–129 (2013)
6. Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)
7. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
8. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)
9. Bauschke, H.H., Noll, D.: On the local convergence of the Douglas–Rachford algorithm. Arch. Math. 102, 589–600 (2014)
10. Bolte, J., Daniilidis, A., Lewis, A.: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17, 1205–1223 (2007)
11. Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18, 556–572 (2007)
12. Borwein, J.M., Li, G., Yao, L.J.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets. SIAM J. Optim. 24, 498–527 (2014)
13. Combettes, P.L., Pesquet, J.-C.: A Douglas–Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Proces. 1, 564–574 (2007)
14. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
15. Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)
16. Gandy, S., Recht, B., Yamada, I.: Tensor completion and low- n -rank tensor recovery via convex optimization. Inverse Probl. 27, 025010 (2011)
17. Gong, P., Zhang, C., Lu, Z., Huang, J., Ye, J.: A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems. In: The 30th International Conference on Machine Learning (ICML 2013)
18. He, B., Yuan, X.: On the O(1 / n) convergence rate of the Douglas–Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)
19. Hesse, R., Luke, D.R.: Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23, 2397–2419 (2013)
20. Hesse, R., Luke, D.R., Neumann, P.: Alternating projections and Douglas–Rachford for sparse affine feasibility. IEEE Trans. Signal. Proces. 62, 4868–4881 (2014)
21. Lewis, A.S., Luke, D.R., Malick, J.: Local convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9, 485–513 (2009)
22. Lewis, A.S., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33, 216–234 (2008)
23. Li, G., Mordukhovich, B.S., Pham, T.S.: New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors. Math. Program. 153, 333–362 (2015)
24. Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
25. Lu, Z., Zhang, Y.: Sparse approximation via penalty decomposition methods. SIAM J. Optim. 23, 2448–2478 (2013)
26. Luke, D.R.: Finding best approximation pairs relative to a convex and a prox-regular set in Hilbert space. SIAM J. Optim. 19, 714–739 (2008)
27. Patrinos, P., Stella, L., Bemporad, A.: Douglas–Rachford splitting: complexity estimates and accelerated variants. In: Proceedings of the 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), pp. 4234–4239. IEEE, Los Angeles, CA (2014)
28. Phan, H.: Linear convergence of the Douglas–Rachford method for two closed sets. Optimization. arXiv:1401.6509
29. Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352, 5231–5249 (2000)
30. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
31. Sun, D., Toh, K.-C., Yang, L.: A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-type of constraints. SIAM J. Optim. 25, 882–915 (2015)
32. 1 / 2 regularization: convergence of iterative half thresholding algorithm. IEEE Trans. Signal Process. 62, 2317–2329 (2014)