Computing a Trust Region Step for a Penalty Function
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We consider the problem of minimizing a quadratic function subject to an ellipsoidal constraint when the matrix involved is the Hessian of a quadratic penalty function (i.e., a function of the form $p(x) = f(x) + \frac{1}{2\mu} c(x)^{T} c(x))$. Most applications of penalty functions require $p(x)$ to be minimized for values of $\mu$ decreasing to zero. In general, as $\mu$ tends to zero the nature of finite precision arithmetic causes a considerable loss of information about the null space of the constraint gradients when $\nabla^{2}p(x)$ is formed. This loss of information renders ordinary trust region Newton's methods unstable and degrades the accuracy of the solution to the trust region problem. The algorithm of More and Sorenson [1983] is modified so as to be more stable and less sensitive to the nature of finite precision arithmetic in this situation. Numerical experiments clearly demonstrate the stability of the proposed algorithm.
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1987-07
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Cornell University
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computer science; technical report
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http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR87-847
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technical report