# A space decomposition scheme for maximum eigenvalue functions and its applications

A space decomposition scheme for maximum eigenvalue functions and its applications 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Methods of Operations Research Springer Journals

# A space decomposition scheme for maximum eigenvalue functions and its applications

, Volume 85 (3) – Mar 18, 2017
38 pages

/lp/springer_journal/a-space-decomposition-scheme-for-maximum-eigenvalue-functions-and-its-Do0yFmHggD
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Operations Research/Decision Theory; Business and Management, general
ISSN
1432-2994
eISSN
1432-5217
D.O.I.
10.1007/s00186-017-0579-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

Mathematical Methods of Operations ResearchSpringer Journals

Published: Mar 18, 2017

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations

Abstract access only

18 million full-text articles

Print

20 pages / month

PDF Discount

20% off