The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications
Ming Z. Huang, Cong Cheng, Yang Li, Zun Quan Xia · Journal of Industrial and Management Optimization · 2019
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.