A space decomposition scheme for maximum eigenvalue functions and its applications

Ming Huang (), Yue Lu, Li Ping Pang and Zun Quan Xia
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Ming Huang: Dalian University of Technology
Yue Lu: Tianjin Normal University
Li Ping Pang: Dalian University of Technology
Zun Quan Xia: Dalian University of Technology

Mathematical Methods of Operations Research, 2017, vol. 85, issue 3, No 6, 453-490

Abstract: Abstract In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the $${\mathcal {U}}$$ U -Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the $${\mathcal {U}}$$ U -Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem.

Keywords: Nonsmooth optimization; Eigenvalue optimization; Matrix-convex; Semidefinite programming; $${\mathcal {VU}}$$ VU -decomposition; $${\mathcal {U}}$$ U -Lagrangian; Smooth manifold; Second-order derivative; Bilinear matrix inequality; 15A18; 52A41; 65K10; 90C25; 90C22; 49J52 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s00186-017-0579-z

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