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Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization

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  • Y. Yang

    (Orbital Sciences Corporation)

Abstract

This paper proposes several globally convergent geometric optimization algorithms on Riemannian manifolds, which extend some existing geometric optimization techniques. Since any set of smooth constraints in the Euclidean space R n (corresponding to constrained optimization) and the R n space itself (corresponding to unconstrained optimization) are both special Riemannian manifolds, and since these algorithms are developed on general Riemannian manifolds, the techniques discussed in this paper provide a uniform framework for constrained and unconstrained optimization problems. Unlike some earlier works, the new algorithms have less restrictions in both convergence results and in practice. For example, global minimization in the one-dimensional search is not required. All the algorithms addressed in this paper are globally convergent. For some special Riemannian manifold other than R n , the new algorithms are very efficient. Convergence rates are obtained. Applications are discussed.

Suggested Citation

  • Y. Yang, 2007. "Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 245-265, February.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:2:d:10.1007_s10957-006-9081-0
    DOI: 10.1007/s10957-006-9081-0
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    References listed on IDEAS

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    1. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
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    Cited by:

    1. X. M. Wang & C. Li & J. C. Yao, 2015. "Subgradient Projection Algorithms for Convex Feasibility on Riemannian Manifolds with Lower Bounded Curvatures," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 202-217, January.
    2. Xiao-bo Li & Nan-jing Huang & Qamrul Hasan Ansari & Jen-Chih Yao, 2019. "Convergence Rate of Descent Method with New Inexact Line-Search on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 830-854, March.
    3. X. M. Wang & J. H. Wang & C. Li, 2023. "Convergence of Inexact Steepest Descent Algorithm for Multiobjective Optimizations on Riemannian Manifolds Without Curvature Constraints," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 187-214, July.

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