Abstract
We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of \(\mathcal {O}\Big (k^{-\frac{1}{2^{d+1}-2}}\Big )\), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of \(\mathcal {O}\Big (\frac{1}{\sqrt{k}}\Big )\).
References
Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1(2), 185–212 (1993)
Bauschke, H.H., Borwein, J.M.: Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79(3), 418–443 (1994)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)
Berman, A.: Cones, Matrices and Mathematical Programming. Lecture Notes in Economics and Mathematical Systems, vol. 79. Springer, Berlin (1973)
Bolte, J., Daniilidis, A., Lewis, A.S.: Generic optimality conditions for semialgebraic convex programs. Math. Oper. Res. 36(1), 55–70 (2011)
Borwein, J., Li, G., Yao, L.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets. SIAM J. Optim. 24(1), 498–527 (2014)
Borwein, J.M.: Proximality and chebyshev sets. Optim. Lett. 1(1), 21–32 (2006)
Borwein, J.M., Moors, W.B.: Stability of closedness of convex cones under linear mappings. J. Convex Anal. 16(3–4), 699–705 (2009)
Borwein, J.M., Moors, W.B.: Stability of closedness of convex cones under linear mappings II. J. Nonlinear Anal. Optim. 1(1), 1–7 (2010)
Borwein, J.M., Wolkowicz, H.: Facial reduction for a cone-convex programming problem. J. Austral. Math. Soc. Ser. A 30(3):369–380 (1980/81)
Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl 6, 688–692 (1965)
Cheung, Y.-L., Schurr, S., Wolkowicz, H.: Preprocessing and regularization for degenerate semidefinite programs. In: Computational and Analytical Mathematics, vol. 50 of Springer Proc. Math. Stat., pp. 251–303. Springer, New York (2013)
Cheung, Y.L.: Preprocessing and Reduction for Semidefinite Programming via Facial Reduction: Theory and Practice. Ph.D. Thesis, University of Waterloo (2013)
Coste, M.: An Introduction to Semialgebraic Geometry. RAAG notes, 78 pages, Institut de Recherche Mathématiques de Rennes (2002)
Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: Generic minimizing behavior in semialgebraic optimization. SIAM J. Optim. 26(1), 513–534 (2016)
Drusvyatskiy, D., Lewis, A.S.: Generic nondegeneracy in convex optimization. Proc. Am. Math. Soc. 139(7), 2519–2527 (2011)
Drusvyatskiy, D., Vavasis, S.A., Wolkowicz, H.: Extreme point inequalities and geometry of the rank sparsity ball. arXiv:1401.4774 (2014)
Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7(6), 1–24 (1967)
Holmes, R.B.: Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics, No. 24. Springer, New York (1975)
Pataki, G.: Strong duality in conic linear programming: facial reduction and extended duals. In: Computational and Analytical Mathematics, vol. 50 of Springer Proc. Math. Stat., pp. 613–634. Springer, New York (2013)
Pataki, G., Tunçel, L.: On the generic properties of convex optimization problems in conic form. Math. Progr. 89(3, Ser. A), 449–457 (2001)
Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)
Sturm, J.F.: Error bounds for linear matrix inequalities. SIAM J. Optim. 10(4), 1228–1248 (2000). (electronic)
Tunçel, L.: Polyhedral and semidefinite programming methods in combinatorial optimization, vol. 27 of Fields Institute Monographs. American Mathematical Society, Providence, RI; Fields Institute for Research in Mathematical Sciences, Toronto (2010)
von Neumann, J.: Functional Operators. II. The Geometry of Orthogonal Spaces. Annals of Mathematics Studies, no. 22. Princeton University Press, Princeton (1950)
Waki, H., Muramatsu, M.: Facial reduction algorithms for conic optimization problems. J. Optim. Theory Appl. 158(1), 188–215 (2013)
Acknowledgments
We thank Levent Tunçel for insightful discussions, and the two anonymous referees for their comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of G. Li was supported by an ARC Future Fellowship. Research of H. Wolkowicz was supported by The Natural Sciences and Engineering Research Council of Canada and AFOSR.
Rights and permissions
About this article
Cite this article
Drusvyatskiy, D., Li, G. & Wolkowicz, H. A note on alternating projections for ill-posed semidefinite feasibility problems. Math. Program. 162, 537–548 (2017). https://doi.org/10.1007/s10107-016-1048-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-016-1048-9
Keywords
- Error bounds
- Regularity
- Alternating projections
- Sublinear convergence
- Linear matrix inequality (LMI)
- Semi-definite program (SDP)