Smooth convex approximation to the maximum eigenvalue function

Chen, Xin, Qi, Houduo, Qi, Liqun and Teo, Kok-Lay (2004) Smooth convex approximation to the maximum eigenvalue function Journal of Global Optimization, 30, (2-3), pp. 253-270. (doi:10.1007/s10898-004-8271-2).


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In this paper, we consider smooth convex approximations to the maximum eigenvalue function. To make it applicable to a wide class of applications, the study is conducted on the composite function of the maximum eigenvalue function and a linear operator mapping Rm to Sn, the space of n-by-n symmetric matrices. The composite function in turn is the natural objective function of minimizing the maximum eigenvalue function over an affine space in Sn. This leads to a sequence of smooth convex minimization problems governed by a smoothing parameter. As the parameter goes to zero, the original problem is recovered. We then develop a computable Hessian formula of the smooth convex functions, matrix representation of the Hessian, and study the regularity conditions which guarantee the nonsingularity of the Hessian matrices. The study on the well-posedness of the smooth convex function leads to a regularization method which is globally convergent.

Item Type: Article
Digital Object Identifier (DOI): doi:10.1007/s10898-004-8271-2
ISSNs: 0925-5001 (print)
Keywords: matrix representation, spectral function, symmetric function, tikhonov regularization
Organisations: Operational Research
ePrint ID: 29650
Date :
Date Event
November 2004Published
Date Deposited: 11 May 2006
Last Modified: 16 Apr 2017 22:21
Further Information:Google Scholar

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