Smooth Convex Approximation to the Maximum Eigenvalue Function

  • Authors:

  • Xin Chen;Houduo Qi;Liqun Qi;Kok-Lay Teo

  • Affiliations:

  • Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 224 Mechanical Engineering Building, mc-244, Urbana, USA 61801;School of Mathematics, University of Southampton, Southampton, S017 1BJ, Great Britain;Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong;Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2004

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Abstract

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 驴m to $${\mathcal{S}}_n $$ , 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 $${\mathcal{S}}_n $$ . 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.