In this paper, we mainly consider optimization problems involving the sum of largest eigenvalues of nonlinear symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, regarded as functions of a symmetric matrix, are not differentiable at those points where they coalesce. The $\mathcal {U}$-Lagrangian theory is applied to the function of the sum of the largest eigenvalues, with convex matrix-valued mappings, which doesn't need to be affine. Some of the results generalize the corresponding conclusions for linear mapping. In the approach, we reformulate the first- and second-order derivatives of ${\mathcal U}$-Lagrangian in the space of decision variables $R^m$ under some mild conditions in terms of $\mathcal{VU}$-space decomposition. We characterize smooth trajectory, along which the function has a second-order expansion. Moreover, an algorithm framework with superlinear convergence is presented. Finally, an application of $\mathcal{VU}$-decomposition derivatives shows that $\mathcal{U}$-Lagrangian possesses proper execution in matrix variable.

中文翻译：

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非线性凸最大特征值之和的空间分解方法及其应用

在本文中，我们主要考虑涉及非线性对称矩阵最大特征值之和的优化问题。对此类问题进行数值分析的困难之一是，将特征值视为对称矩阵的函数，在它们合并的那些点无法区分。$ \ mathcal {U} $-Lagrangian理论应用于具有凸矩阵值映射的最大特征值之和的函数，不需要仿射。一些结果概括了线性映射的相应结论。在该方法中，我们在某些较温和的条件下，根据$ \ mathcal {VU} $-，在决策变量$ R ^ m $的空间中重新制定了$ {\ mathcal U} $-Lagrangian的一阶和二阶导数。空间分解。我们描绘出光滑的轨迹，沿该函数具有二阶展开。此外，提出了一种具有超线性收敛的算法框架。最后，应用$ \ mathcal {VU} $-分解导数表明$ \ mathcal {U} $-Lagrangian在矩阵变量中具有适当的执行力。