mlsanityData to 14 Oct 2026

A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints

arXiv 2511.03443 ↗First posted 5 Nov 2025math.OC cs.LG stat.MLcited by 0

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Abstract (arXiv)

In this paper, we investigate optimization problems with nonnegative and orthogonal constraints, where any feasible matrix of size n × p exhibits a sparsity pattern such that each row accommodates at most one nonzero entry. Our analysis demonstrates that, by fixing the support set, the global solution of the minimization subproblem for the proximal linearization of the objective function can be computed in closed form with at most n nonzero entries. Exploiting this structural property offers a powerful avenue for dramatically enhancing computational efficiency. Guided by this insight, we propose a support-set algorithm preserving strictly the feasibility of iterates. A central ingredient is a strategically devised update scheme for support sets that adjusts the placement of nonzero entries. We establish the convergence of the support-set algorithm to a first-order stationary point, and show that its iteration complexity required to reach an ε-approximate first-order stationary point is O (ε-2). Numerical results are strongly in favor of our algorithm in real-world applications, including nonnegative PCA, clustering, and community detection.

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References · 17

  1. 2019Simple algorithms for optimization on Riemannian manifolds with constraintscited by 14
  2. 2021Decentralized Optimization Over the Stiefel Manifold by an Approximate Augmented Lagrangian Functioncited by 11
  3. 2019Clustering by Orthogonal NMF Model and Non-Convex Penalty Optimizationcited by 9
  4. 2019An exact penalty approach for optimization with nonnegative orthogonality constraintscited by 7
  5. 2021A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifoldscited by 6
  6. 2020Multipliers Correction Methods for Optimization Problems over the Stiefel Manifoldcited by 4
  7. 2024A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifoldscited by 4
  8. 2025An Exact Penalty Approach for Equality Constrained Optimization over a Convex Setcited by 4
  9. 2016Orthogonal symmetric non-negative matrix factorization under the stochastic block modelcited by 3
  10. 2021A Communication-Efficient and Privacy-Aware Distributed Algorithm for Sparse PCAcited by 3
  11. 2021Error bound and exact penalty method for optimization problems with nonnegative orthogonal constraintcited by 3
  12. 2022Tight Error Bounds for the Sign-Constrained Stiefel Manifoldcited by 3
5 more
  1. 2024Unsupervised Feature Selection via Nonnegative Orthogonal Constrained Regularized Minimizationcited by 3
  2. 2025On exploration of an interior mirror descent flow for stochastic nonconvex constrained problemcited by 3
  3. 2016Lp-norm regularization algorithms for optimization over permutation matricescited by 2
  4. 2020Seeking Consensus on Subspaces in Federated Principal Component Analysiscited by 2
  5. 2017A Sparse Completely Positive Relaxation of the Modularity Maximization for Community Detectioncited by 1

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