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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.

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