The $$\omega $$ ω -condition number: applications to preconditioning and low rank generalized Jacobian updating

Woosuk L. Jung (), David Torregrosa-Belén () and Henry Wolkowicz ()
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Woosuk L. Jung: University of Waterloo
David Torregrosa-Belén: Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile
Henry Wolkowicz: University of Waterloo

Computational Optimization and Applications, 2025, vol. 91, issue 1, No 8, 235-282

Abstract: Abstract Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g., in quasi-Newton methods. We study a nonclassic matrix condition number, the $$\omega $$ ω -condition number, $$\omega $$ ω for short. $$\omega $$ ω is the ratio of: the arithmetic and geometric means of the singular values, rather than the largest and smallest for the classical $$\kappa $$ κ -condition number. The simple functions in $$\omega $$ ω allow one to exploit first order optimality conditions. We use this fact to derive explicit formulae for (i) $$\omega $$ ω -optimal low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) $$\omega $$ ω -optimal preconditioners of special structure for iterative methods for linear systems. In the latter context, we analyze the benefits of $$\omega $$ ω for (a) improving the clustering of eigenvalues; (b) reducing the number of iterations; and (c) estimating the actual condition of a linear system. Moreover we show strong theoretical connections between the $$\omega $$ ω -optimal preconditioners and incomplete Cholesky factorizations, and highlight the misleading effects arising from the inverse invariance of $$\kappa $$ κ . Our results confirm the efficacy of using the $$\omega $$ ω -condition number compared to the $$\kappa $$ κ -condition number.

Keywords: $$\kappa; \omega; \omega ^{-2}$$ κ; ω; ω - 2 -Condition numbers; Preconditioning; Generalized Jacobian; Iterative methods; Clustering of eigenvalues; 15A12; 65F35; 65F08; 65G50; 49J52; 49K10; 90C32 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-025-00669-w

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