A Curvilinear Search Using Tridiagonal Secant Updates for Unconstrained Optimization
RICE UNIV HOUSTON TX DEPT OF MATHEMATICAL SCIENCES
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The idea of doing a curvilinear search along the Levenberg-Marquardt path smu -gH muI always has been appealing, but the cost of solving a linear system for each trial value of the parameter mu has discouraged its implementation. In this paper, an algorithm for searching along a path which includes smu is studied. The algorithm uses a special inexpensive QTcQ-superscript-T to QTQ-superscript-T Hessian update which trivializes the linear algebra required to compute smu. This update is based on earlier work of Dennis-Marwil and Martinez on least-change secant updates of matrix factors. The new algorithm is shown to be local and q-superlinearily convergent to stationary points, and to be globally q-superlinearily convergent for quasi-convex functions. Computational tests are given that show the new algorithm to be robust and efficient.
- Theoretical Mathematics