Abstract
Properties of the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix are derived. Such a sum is convex but not necessarily everywhere differentiable. A convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the matrix is held constant. An implementation of this procedure is described and numerical results are included.
Minimization problems of this kind arose in graph partitioning studies [8]. Use of existing procedures for minimizing required either a strategy for selecting, at each stage, a direction of search from the subdifferential and an appropriate step along the direction chosen [10,13] or computationally feasible characterizations of certain enlargements of subdifferentials [1,6] neither of which could be easily determined for the given problem. The arguments use results from eigenelement analysis and from optimization theory.
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© 1975 The Mathematical Programming Society
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Cullum, J., Donath, W.E., Wolfe, P. (1975). The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. In: Balinski, M.L., Wolfe, P. (eds) Nondifferentiable Optimization. Mathematical Programming Studies, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120698
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DOI: https://doi.org/10.1007/BFb0120698
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