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Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone

Author

Listed:
  • H. H. Bauschke

    (University of Guelph)

  • S. G. Kruk

    (Oakland University)

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.

Suggested Citation

  • H. H. Bauschke & S. G. Kruk, 2004. "Reflection-Projection Method for Convex Feasibility Problems with an Obtuse Cone," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 503-531, March.
    • Handle: RePEc:spr:joptap:v:120:y:2004:i:3:d:10.1023_b:jota.0000025708.31430.22
    DOI: 10.1023/B:JOTA.0000025708.31430.22
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    References listed on IDEAS

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    1. J. L. Goffin, 1980. "The Relaxation Method for Solving Systems of Linear Inequalities," Mathematics of Operations Research, INFORMS, vol. 5(3), pages 388-414, August.
    2. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
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    Cited by:

    1. Meng Wen & Jigen Peng & Yuchao Tang, 2015. "A Cyclic and Simultaneous Iterative Method for Solving the Multiple-Sets Split Feasibility Problem," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 844-860, September.
    2. Minh N. Dao, & Hung M. Phan, 2019. "Linear Convergence of Projection Algorithms," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 715-738, May.
    3. Heinz H. Bauschke & Caifang Wang & Xianfu Wang & Jia Xu, 2015. "On the Finite Convergence of a Projected Cutter Method," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 901-916, June.
    4. Jinhua Wang & Yaohua Hu & Carisa Kwok Wai Yu & Xiaojun Zhuang, 2019. "A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 520-534, November.
    5. Heinz H. Bauschke & Minh N. Dao & Dominikus Noll & Hung M. Phan, 2016. "On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces," Journal of Global Optimization, Springer, vol. 65(2), pages 329-349, June.

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