A Decomposition Algorithm for the Sums of the Largest Eigenvalues. (8th September 2020) - Huang, Ming; Lu, Yue; Yuan, Jin Long; Li, Yang
Abstract: In this article, we consider optimization problems in which the sums of the largest eigenvalues of symmetric matrices are involved. Considered as functions of a symmetric matrix, the eigenvalues are not smooth once the multiplicity of the function is not single; this brings some difficulties to solve. For this, the function of the sums of the largest eigenvalues with affine matrix-valued mappings is handled through the application of the U -Lagrangian theory. Such theory extends the corresponding conclusions for the largest eigenvalue function in the literature. Inspired V U -space decomposition, the first- and second-order derivatives of U -Lagrangian in the space of decision variables R m are proposed when some regular condition is satisfied. Under this condition, we can use the vectors of V -space to generate an implicit function, from which a smooth trajectory tangent to U can be defined. Moreover, an algorithm framework with superlinear convergence can be presented. Finally, we provide an application about arbitrary eigenvalue which is usually a class of DC functions to verify the validity of our approach.