Breaking the Barrier of Self-Concordant Barriers: Faster Interior Point Methods for M-Matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918004163346432 |
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| author | Vladu, Adrian |
| author_facet | Vladu, Adrian |
| contents | We study two fundamental optimization problems: (1) scaling a symmetric positive definite matrix by a positive diagonal matrix so that the resulting matrix has row and column sums equal to 1; and (2) minimizing a quadratic function subject to hard non-negativity constraints. Both problems lend themselves to efficient algorithms based on interior point methods (IPMs). For general instances, standard self-concordance theory places a limit on the iteration complexity of these methods at $\widetilde{O}\left(n^{1/2}\right)$, where $n$ denotes the matrix dimension. We show via an amortized analysis that, when the input matrix is an M-matrix, an IPM with adaptive step sizes solves both problems in only $\widetilde{O}\left(n^{1/3}\right)$ iterations. As a corollary, using fast Laplacian solvers, we obtain an $\ell_{2}$ flow diffusion algorithm with depth $\widetilde{O}\left(n^{1/3}\right)$ and work $\widetilde{O}$$\left(n^{1/3}\cdot\text{nnz}\right)$. This result marks a significant instance in which a standard log-barrier IPM permits provably fewer than $Θ\left(n^{1/2}\right)$ iterations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Breaking the Barrier of Self-Concordant Barriers: Faster Interior Point Methods for M-Matrices Vladu, Adrian Data Structures and Algorithms Optimization and Control We study two fundamental optimization problems: (1) scaling a symmetric positive definite matrix by a positive diagonal matrix so that the resulting matrix has row and column sums equal to 1; and (2) minimizing a quadratic function subject to hard non-negativity constraints. Both problems lend themselves to efficient algorithms based on interior point methods (IPMs). For general instances, standard self-concordance theory places a limit on the iteration complexity of these methods at $\widetilde{O}\left(n^{1/2}\right)$, where $n$ denotes the matrix dimension. We show via an amortized analysis that, when the input matrix is an M-matrix, an IPM with adaptive step sizes solves both problems in only $\widetilde{O}\left(n^{1/3}\right)$ iterations. As a corollary, using fast Laplacian solvers, we obtain an $\ell_{2}$ flow diffusion algorithm with depth $\widetilde{O}\left(n^{1/3}\right)$ and work $\widetilde{O}$$\left(n^{1/3}\cdot\text{nnz}\right)$. This result marks a significant instance in which a standard log-barrier IPM permits provably fewer than $Θ\left(n^{1/2}\right)$ iterations. |
| title | Breaking the Barrier of Self-Concordant Barriers: Faster Interior Point Methods for M-Matrices |
| topic | Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2504.20619 |