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Large-scale optimization of eigenvalues. (English) Zbl 0757.65072
The paper is concerned with a large scale optimization problem involving eigenvalues of a symmetric \(n\times n\) matrix \(A(x)\), where \(A(x)\) depends smoothly on a vector of parameters \(x\in\mathbb{R}^ m\). This is a nonsmooth optimization problem due to the nondifferentiability of the eigenvalues of \(A(x)\) at points \(x\) where they coalesce.
The proposed algorithm uses a dual matrix formulation of the optimality conditions to fully exploit the nonsmooth problem structure. The eigenvalues of the dual matrix are also used for sensitivity analysis of optimal solutions.
A successive partial linear programming method is implemented for large \(m\) (\(m>40\)) and it is shown how to efficiently compute the eigenvalues of the matrix iterates generated by the optimization algorithm when \(n\) is large. Numerical results of the algorithm are discussed in detail.

65K05 Numerical mathematical programming methods
90C26 Nonconvex programming, global optimization
90C06 Large-scale problems in mathematical programming
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