{"ID":3053322,"CreatedAt":"2026-06-04T04:41:36.695875263Z","UpdatedAt":"2026-06-06T01:20:22.681628739Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04348","arxiv_id":"2606.04348","title":"A Perturbed q-Tsallis Self-Concordant Barrier for Spectrally Robust Semidefinite Programming","abstract":"We introduce and analyse a perturbed $q$-Tsallis barrier for semidefinite programming (SDP), defined as a spectral perturbation of the classical log-det barrier on the cone of positive definite matrices. The barrier introduces eigenvalue-adaptive stiffening through a Tsallis-type matrix-power term controlled by parameters $q\u003e1$ and $η\\geq0$. Our main theoretical contribution is a sharp characterisation of the differential self-concordance regime of the barrier. We prove that the barrier is differentially self-concordant on the interior of the positive semidefinite cone for all $η\\geq0$ if and only if $q\\in(1,2]$, establishing the exact threshold at $q=2$. For $q\u003e2$, uniform self-concordance fails globally, although local sufficient conditions remain valid on compact spectral domains. On compact feasible sets, the effective local barrier parameter remains $O(n)$, preserving the same asymptotic iteration complexity class as the classical log-det barrier. We further establish a spectral robustness result showing that the sensitivity of the central path to perturbations is selectively damped in small-eigenvalue directions according to the scaling $κ(X^*)^{-(q-1)}$, where $κ(X^*)$ denotes the spectral condition number. This yields improved robustness relative to the classical log-det barrier for ill-conditioned SDP solutions. Finally, we develop a Mehrotra-type primal--dual predictor--corrector interior-point method equipped with a Lanczos-based Krylov kernel for evaluating matrix powers efficiently. Numerical experiments validate the theoretical predictions and demonstrate improved robustness together with significant computational acceleration.","short_abstract":"We introduce and analyse a perturbed $q$-Tsallis barrier for semidefinite programming (SDP), defined as a spectral perturbation of the classical log-det barrier on the cone of positive definite matrices. The barrier introduces eigenvalue-adaptive stiffening through a Tsallis-type matrix-power term controlled by paramet...","url_abs":"https://arxiv.org/abs/2606.04348","url_pdf":"https://arxiv.org/pdf/2606.04348v1","authors":"[\"Sergio Assuncao Monteiro\",\"Fabricio Alves Barbosa da Silva\"]","published":"2026-06-03T02:04:59Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.NA\"]","methods":"[]","has_code":false}