Consensus optimization on manifolds

Alain Sarlette; Rodolphe Sepulchre

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arXiv · 0811.4275

Consensus optimization on manifolds

Alain Sarlette·Rodolphe Sepulchre

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Abstract

The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Gr(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle.

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Alain Sarlette, Rodolphe Sepulchre. 2008-11-26. Consensus optimization on manifolds. https://arxiv.org/abs/0811.4275

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