Abstract
The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets in Euclidean space. This problem is of fundamental importance in the mathematical and physical sciences, and it can be solved algorithmically by the classical method of cyclic projections.
In this paper, the case where one of the constraints is an obtuse cone is considered. Because the nonnegative orthant as well as the set of positive-semidefinite symmetric matrices form obtuse cones, we cover a large and substantial class of feasibility problems. Motivated by numerical experiments, the method of reflection-projection is proposed: it modifies the method of cyclic projections in that it replaces the projection onto the obtuse cone by the corresponding reflection.
This new method is not covered by the standard frameworks of projection algorithms because of the reflection. The main result states that the method does converge to a solution whenever the underlying convex feasibility problem is consistent. As prototypical applications, we discuss in detail the implementation of two-set feasibility problems aiming to find a nonnegative [resp. positive semidefinite] solution to linear constraints in ℝn [resp. in \(\mathbb{S}^n \), the space of symmetric n×n matrices] and we report on numerical experiments. The behavior of the method for two inconsistent constraints is analyzed as well.
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Bauschke, H. H., Projection Algorithms and Monotone Operators, PhD Thesis, Simon Fraser University, 1996. Available at www.cecm.sfu.ca/preprints/1996pp.html as 96:080.
Bauschke, H. H., and Borwein, J. M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Review, 38, 367-426, 1996.
Censor, Y., and Zenios, S. A., Parallel Optimization, Oxford University Press, Oxford, UK, 1997.
Combettes, P. L., Hilbertian Convex Feasibility Problem: Convergence of Projection Methods, Applied Mathematics and Optimization, 35, 311-330, 1997.
Crombez, G., Parallel Algorithms for Finding Common Fixed Points of Paracontractions, Numerical Functional Analysis and Optimization, 23, 47-59, 2002.
Kiwiel, K. C., and Łopuch, B., Surrogate Projection Methods for Finding Fixed Points of Firmly Nonexpansive Mappings, SIAM Journal of Optimization, 7, 1084-1102, 1997.
Motzkin, T. S., and Schoenberg, I. J., The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, 6, 393-404, 1954.
Cimmino, G., Calcolo Approssimato per le Soluzioni dei Sistemi di Equazioni Lineari, La Ricerca Scientifica ed il Progresso Tecnico nella Economia Nazionale (Roma), 1, 326-333, 1938.
Goebel, K., and Kirk, W. A., Topics in Metric Fixed-Point Theory, Cambridge University Press, Cambridge, UK, 1990.
Zarantonello, E. H., Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Academic Press, New York, NY, 237-424, 1971.
Moreau, J. J., Décomposition Orthogonale d'un Espace Hilbertien selon Deux Cônes Mutuellement Polaires, Comptes Rendus des Séances de l'Académie des Sciences, Paris, Séries A-B, 255, 238-240, 1962.
Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Goffin, J. L., The Relaxation Method for Solving Systems of Linear Inequalities, Mathematics of Operations Research, 5, 388-414, 1980.
Todd, M. J., Some Remarks on the Relaxation Method for Linear Inequalities, Technical Report 419, School of Operations Research and Industrial Engineering, Cornell University, 1979.
Cegielski, A., Projection onto an Acute Cone and Convex Feasibility Problem, System Modelling and Optimization (Compiègne, 1993); Lecture Notes in Control and Information Sciences, J. Henry and J. P. Yvon, Springer Verlag, London, UK, 197, 187-194, 1994.
Cegielski, A., A Method of Projection onto an Acute Cone with Level Control in Convex Minimization, Mathematical Programming, 85, 469-490, 1999.
Cegielski, A., Obtuse Cones and Gram Matrices with Nonnegative Inverse, Linear Algebra and Its Applications, 335, 167-181, 2001.
Cegielski, A., and Dylewski, R., Residual Selection in a Projection Method for Convex Minimization Problems, Optimization, 52, 211-220, 2003.
Kiwiel, K. C., The Efficiency of Subgradient Projection Methods for Convex Optimization, Part 2: Implementations and Extensions, SIAM Journal on Control and Optimization, 34, 677-697, 1996.
Kiwiel, K. C., Monotone Gram Matrices and Deepest Surrogate Inequalities in Accelerated Relaxation Methods for Convex Feasibility Problems, Linear Algebra and Its Applications, 252, 27-33, 1997.
Güler, O., Barrier Functions in Interior-Point Methods, Mathematics of Operations Research, 21, 860-885, 1996.
Nesterov, Y. E., and Todd, M. J., Self-Scaled Barriers and Interior-Point Methods for Convex Programming, Mathematics of Operations Research, 22, 1-42, 1997.
Lobo, M. S., Vandenberghe, L., Boyd, S., and Lebret, H., Applications of Second-Order Cone Programming, Linear Algebra and Its Applications, 284, 193-228, 1998.
Bruck, R. E., and Reich, S. Nonexpansive Projections and Resolvents of Accretive Operators in Banach Spaces, Houston Journal of Mathematics, 3, 459-470, 1977.
De Pierro, A. R., and Iusem, A. N., On the Asymptotic Behavior of Some Alternate Smoothing Series Expansion Iterative Methods, Linear Algebra and Its Applications, 130, 3-24, 1990.
Bauschke, H. H., Projection Algorithms: Results and Open Problems, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000); D. Butnariu, Y. Censor, and S. Reich, Elsevier, Amsterdam, Holland, 11-22, 2001.
Combettes, P. L., Quasi-Fejérian Analysis of Some Optimization Algorithms, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000); D. Butnariu, Y. Censor, and S. Reich, Elsevier, Amsterdam, Holland, 115-152, 2001.
Groetsch, C. W., Generalized Inverses of Linear Operators: Representation and Approximation, Dekker, New York, NY, 1977.
Deutsch, F., Best Approximation in Inner Product Spaces, Springer, New York, NY, 2001.
Golub, G. H., and Van Loan, C. F., Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 1996.
Kruk, S., Implementation of the Algorithms Discussed in This Manuscript, www.oakland.edu/~kruk/research/ProjRefl.
Horn, R. A., and Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
Borwein, J. M., and Lewis, A. S., Convex Analysis and Nonlinear Optimization, Springer, New York, NY, 2000.
Wolkowicz, H., Saigal, R., and Vandenberghe, L., Editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, International Series in Operations Research and Management Science, Kluwer, Norwell, Massachusetts, 27, 2000.
Vandenberghe, L., and Boyd, S., Semidefinite Programming, SIAM Review, 38, 49-95, 1996.
Parlett, B. N., The Symmetric Eigenvalue Problem, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1998 (Corrected Reprint of the 1980 Original).
Stewart, G. W., Matrix Algorithms, 2: Eigensystems, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2001.
Bauschke, H. H., Borwein, J. M., and Lewis, A. S., The Method of Cyclic Projections for Closed Convex Sets in Hilbert Space, Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995); Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 204, 1-38, 1997.
Cheney, W., and Goldstein, A. A., Proximity Maps for Convex Sets, Proceedings of the American Mathematical Society, 10, 448-450, 1959.
Bauschke, H. H., and Borwein, J. M., Dykstra's Alternating Projection Algorithm for Two Sets, Journal of Approximation Theory, 79, 418-443, 1994.
de Klerk, E., Roos, C., and Terlaky, T., Infeasible-Start Semidefinite Programming Algorithms via Self-Dual Embeddings, Topics in Semidefinite and Interior-Point Methods (Toronto, 1996); P. M. Pardalos and H. Wolkowicz, American Mathematical Society, Providence, Rhode Island, 215-236, 1998.
Luo, Z. Q., Sturm, J. F., and Zhang, S., Conic Convex Programming and Self-Dual Embedding, Optimization Methods and Software, 14, 169-218, 2000.
Strang, G., Linear Algebra and Its Applications, Academic Press, New York, NY, 1976.
Kruk, S., High-Accuracy Algorithms for the Solutions of Linear Programs, University of Waterloo, PhD Thesis, 2001.
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Bauschke, H.H., Kruk, S.G. Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone. Journal of Optimization Theory and Applications 120, 503–531 (2004). https://doi.org/10.1023/B:JOTA.0000025708.31430.22
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DOI: https://doi.org/10.1023/B:JOTA.0000025708.31430.22