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Riemannian conditional gradient methods for composite optimization problems
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DOI：10.1007/s40314-025-03496-1.png)
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Abstract
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In this paper, we propose Riemannian conditional gradient methods for minimizing composite functions, i.e., those that can be expressed as the sum of a smooth function and a retraction-based convex function. We analyze the convergence of the proposed algorithms, utilizing three types of step-size strategies: adaptive, diminishing, and those based on the Armijo condition. We establish the convergence rate of $$\mathcal {O}(1/k)$$ for the adaptive and diminishing step sizes, where $$k$$ denotes the number of iterations. Additionally, we derive an iteration complexity of $$\mathcal {O}(1/\epsilon ^2)$$ for the Armijo step-size strategy to achieve $$\epsilon $$ -optimality, where $$\epsilon $$ is the optimality tolerance. Finally, the effectiveness of our algorithms is validated through some numerical experiments performed on the sphere and Stiefel manifolds.
Keywords:
Conditional gradient methods
Riemannian manifolds
Composite optimization problems
Frank-Wolfe methods
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