分数规划的二次变换与Schur补之间的联系
Connections Between Quadratic Transform for Fractional Programming and Schur Complement
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中文总结 AI 辅助
本文揭示了分数规划二次变换与Schur补的深层联系，并推广至广义矩阵逆，应用于高斯广播信道和容量问题，恢复上行-下行对偶关系。
中文摘要 AI 辅助
本文表明，用于求解分数规划（FP）问题的二次变换技术与矩阵分析中的Schur补技术之间存在紧密联系。我们证明二次变换技术与Schur补的两个方面相关：（i）半正定性的线性矩阵不等式（LMI）条件，以及（ii）矩阵行列式公式。具体而言，我们建立了二次变换与Schur补LMI条件相互蕴含的关系。这一联系使我们能够对二次变换中的辅助变量提供新的解释，并能够重新推导Schur补行列式公式。此外，这一联系还导致了FP中二次变换和Schur补LMI的推广，这些推广可以容纳广义矩阵逆。作为信息论中的一个应用，我们将广义FP框架应用于高斯向量广播信道和容量问题的最小不利噪声极小极大公式。当最小不利噪声协方差是奇异的时候，基于矩阵逆的Karush-Kuhn-Tucker（KKT）分析需要对信道的输入和输出空间进行仔细分析。我们证明，使用广义FP，奇异矩阵分数的辅助变量表示直接产生互易多址信道，并恢复了和容量的上行-下行对偶关系。
英文摘要
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.
发表机构
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
- Shenzhen University（深圳大学）
- University of Toronto（多伦多大学）
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