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In this paper, we focus on the decentralized optimization problem over the Stiefel manifold, which is defined on a connected network of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> agents. The objective is an average of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> local functions, and each function is privately held by an agent and encodes its data. The agents can only communicate with their neighbors in a collaborative effort to solve this problem. In existing methods, multiple rounds of communications are required to guarantee the convergence, giving rise to high communication costs. In contrast, this paper proposes a decentralized algorithm, called DESTINY, which only invokes a single round of communications per iteration. DESTINY combines gradient tracking techniques with a novel approximate augmented Lagrangian function. The global convergence to stationary points is rigorously established. Comprehensive numerical experiments demonstrate that DESTINY has a strong potential to deliver a cutting-edge performance in solving a variety of testing problems.

Decentralized Optimization Over the Stiefel Manifold by an Approximate Augmented Lagrangian Function