A Decomposition Algorithm for the Sums of the Largest Eigenvalues

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 VU-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.