A Decomposition Algorithm for the Sums of the Largest Eigenvalues. (8th September 2020)

• Record Type: Journal Article
• Authors: Huang, Ming; Lu, Yue; Yuan, Jin Long; Li, Yang
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 16(2020)
• Page Start: 1936
• Page End: 1969
• Publication Date: 2020-09-08
• Subjects: VU-decomposition -- U-Lagrangian -- nonsmooth optimization -- second-order derivative -- smooth track -- sum of eigenvalues; 52A41 -- 65K10 -- 49J52 -- 15A18; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2020.1813758
• Languages: English
• ISSNs: 0163-0563

Abstract

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.