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Stability analysis of conically perturbed linearly constrained least-squares problems by optimizing the regularized trajectories

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Abstract

This paper studies linearly constrained least-square optimization problems in Hilbert spaces for which the KKT system is not necessarily available to analyze and compute the solution. The primary objective is to develop new qualitative and quantitative stability estimates for the regularization error in the conical regularization approach. To attain this goal, we associate the notion of stability with the solvability of some scalar and vector optimization problems defined in terms of the regularized trajectory on the domain space and the regularized state trajectory on the constraint space. We analyze three optimization formulations. The first formulation minimizes a scalar objective function over the regularized trajectory. The second formulation consists of vector optimizing the regularized trajectory on the domain space for a specific Bishop–Phelps cone. The third formulation results in vector optimizing the regularized state trajectory for the constraint cone. We provide numerical examples to illustrate the efficacy of the developed framework.

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Acknowledgements

The research of Akhtar Khan is supported by the National Science Foundation under Award No. 1720067. Miguel Sama is supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE) and Grant No. 2021-MAT11 (ETSI Industriales, UNED).

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Authors and Affiliations

  1. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    A. Khan

  2. Departamento de Matemática Aplicada, E.T.S.I.I. Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, 28040, Madrid, Spain

    M. Sama

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Correspondence to M. Sama.

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Khan, A., Sama, M. Stability analysis of conically perturbed linearly constrained least-squares problems by optimizing the regularized trajectories. Optim Lett 15, 2127–2145 (2021). https://doi.org/10.1007/s11590-021-01722-3

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