Abstract
The testing of optimization algorithms requires the running of problems with ill-conditioned Hessians. For constrained problems, it is the projection of the Hessian onto the space determined by the active constraints that must be ill conditioned. In this note it is argued that unless the Hessian and the constraints are constructed together, the constrained Hessian is likely to be well conditioned. The approach is to examine the effects of random constraints on a singular Hessian.
References
P.R. Halmos,Measure theory (Van Nostrand, Princeton, NJ, 1950).
A.S. Householder,The theory of matrices in numerical analysis (Blaisdell, New York, 1964).
G.W. Stewart, “The efficient generation of random orthogonal matrices with an application to condition estimators”,SIAM Journal on Numerical Analysis 17 (1980) 403–409.
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This work was supported in part by the Office of Naval Research under Contract No. N00014-76-C-0391.
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Stewart, G.W. Constrained definite Hessians tend to be well conditioned. Mathematical Programming 21, 235–238 (1981). https://doi.org/10.1007/BF01584245
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DOI: https://doi.org/10.1007/BF01584245