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Connections Between Quadratic Transform for Fractional Programming and Schur Complement — arXiv:2609.21730

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.21730
License (as archived): CC0-1.0
Archived: 2026-10-10T02:20:08.563Z
SHA-256 of the archived text: 4aca5e1bb8ecb0df35871555fc4435ba9344275ed0f5374bcdd6e8553fcfd5f7
Stable URL of this page: https://metraipolis.com/documento/dc_ba080718e014

How to cite

Connections Between Quadratic Transform for Fractional Programming and Schur Complement — arXiv:2609.21730 — arXiv (en), https://arxiv.org/abs/2609.21730 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ba080718e014 (Metraipolis archive dc_ba080718e014, sha256 4aca5e1bb8ecb0df35871555fc4435ba9344275ed0f5374bcdd6e8553fcfd5f7)

BibTeX

@misc{dcba080718e014,
  title = {Connections Between Quadratic Transform for Fractional Programming and Schur Complement — arXiv:2609.21730},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.21730},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ba080718e014; sha256 4aca5e1bb8ecb0df35871555fc4435ba9344275ed0f5374bcdd6e8553fcfd5f7},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.21730. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

This paper shows that there are intimate connections between the quadratic transform technique for solving fractional programming (FP) problems and the Schur-complement technique in matrix analysis. We demonstrate that the quadratic transform technique is related to two aspects of the Schur complement: (i) the linear matrix inequality (LMI) condition for positive semidefiniteness and (ii) the matrix determinant formula. Specifically, we establish that the quadratic transform and the Schur-complement LMI condition imply each other. This connection allows us to provide new interpretations of the auxiliary variable in the quadratic transform, and it allows us to rederive the Schur-complement determinant formula. Furthermore, this connection leads to generalizations of the quadratic transform in FP and the Schur-complement LMI that can accommodate generalized matrix inverse. As an application in information theory, we apply the generalized FP framework to the least-favorable-noise minimax formulation of the Gaussian vector broadcast channel sum capacity problem. When the least-favorable noise covariance is singular, matrix-inverse-based Karush-Kuhn-Tucker (KKT) analysis would require a careful analysis of the input and output spaces of the channel. We show using generalized FP that an auxiliary-variable representation of the singular matrix fraction directly yields the reciprocal multiple-access channel and recovers the uplink-downlink duality relation for sum capacity.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-21 in math.IT).
Categorie arXiv: Information Theory (cs.IT).
Pagina del lavoro: https://arxiv.org/abs/2609.21730 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-10T02:04:28.749Z dalla pagina https://arxiv.org/catchup/math.IT/2026-09-21?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).