Abstract
Hessian matrices play a key role in optimization. Knowledge of their behavior is useful both in giving insight into optimization problems and in designing algorithms to solve them. In this paper, analytical expressions are obtained for the eigenvalues and eigenvectors at the intermediate minima of barrier and penalty functions. This in turn leads to an analytical expression for the inverse of the Hessian matrix (it is singular) at the solution.
References
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Murray, W.,Ill-Conditioning in Barrier and Penalty Functions Arising in Constrained Nonlinear Programming, Paper Presented at the Sixth Symposium on Mathematical Programming, Princeton, New Jersey, 1967.
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Murray, W.,Constrained Optimization, National Physical Laboratory, Mathematics Division, Report No. 79, 1969.
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Communicated by A. Milele
The author acknowledges the programming assistance of D. Dennis. This work has been carried out at the National Physical Laboratory, Teddington, Middlesex, England.
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Murray, W. Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions. J Optim Theory Appl 7, 189–196 (1971). https://doi.org/10.1007/BF00932477
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DOI: https://doi.org/10.1007/BF00932477