加权幂平均二次型的锐二次上界
Sharp Quadratic Majorants of Weighted Power Means of Quadratic Forms
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中文总结 AI 辅助
本文研究半正定二次型凹加权幂平均的锐二次上界，提出下包络凸性条件保证锐性，并给出线性矩阵不等式验证及变分公式，涵盖元引理等特例。
中文摘要 AI 辅助
我们研究何时半正定二次型的凹加权幂平均允许锐二次上界。出发点是标量事实：每个这样的平均值是其支撑仿射上界的下包络。当标量变量被二次型替换后，所得界在二次上界中可能不再是最优的。我们证明，在联合数值范围的下包络凸性条件下，锐性得以保持。在这种情况下，涉及二次型幂平均的非线性不等式由单个线性矩阵不等式无间隙地验证。该条件弱于完全增广范围的凸性，并且对两个二次型自动成立。这为整个凹幂平均尺度提供了一个锐的彼得-保罗型上界，包括元引理和加权几何平均作为特例。我们还推导了单位球面上二次型幂平均和的极值的变分公式；在相同的下包络条件下，最大化问题变为精确的凸特征值最小化。一个有限时域控制应用说明了该证书如何被建设性地使用。
英文摘要
We study when concave weighted power means of positive semidefinite quadratic forms admit sharp quadratic upper bounds. The starting point is the scalar fact that each such mean is the lower envelope of its supporting affine majorants. After the scalar variables are replaced by quadratic forms, the resulting bounds may cease to be extremal among quadratic majorants. We prove that sharpness is preserved under a lower-envelope convexity condition on the joint numerical range. In that case, a nonlinear inequality involving power means of quadratic forms is certified, with no gap, by a single linear matrix inequality. The condition is weaker than convexity of the full augmented range and holds automatically for two quadratic forms. This yields a sharp Peter-Paul-type majorant for the whole concave power-mean scale, including Yuan's lemma and the weighted geometric mean as special cases. We also derive variational formulas for the extrema of sums of power means of quadratic forms on the unit sphere; under the same lower-envelope condition, the maximization becomes an exact convex eigenvalue minimization. A finite-horizon control application illustrates how the certificate can be used constructively.
发表机构
- University of Sheffield（谢菲尔德大学）
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