Abstract
On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group \(\mathbf{O}(n)\) by using only Euclidean coordinates on \({\mathbb {R}}^{n^2}\). An optimization problem on \(\mathbf{SO}(3)\) is completely carried out. Its applications to nonlinear stability problems are also analyzed.
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Notes
In the formulas for the restricted Hessian matrices of the constraint functions we have used the fact that \(\tilde{\mathbf{x}}\in \mathcal {O}(3)\).
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Acknowledgments
This work was supported by the grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, under the Romania-Cyprus bilateral cooperation programme (module III), Project Number 760/2014. We are also thankful to Ioan Casu for his help with Maple programming.
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Communicated by Paul Newton.
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Birtea, P., Comănescu, D. Hessian Operators on Constraint Manifolds. J Nonlinear Sci 25, 1285–1305 (2015). https://doi.org/10.1007/s00332-015-9256-7
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DOI: https://doi.org/10.1007/s00332-015-9256-7