fundamental-engineering-optimization-methods.pdf

The application of newtons method relies on the

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The application of Newton’s method relies on the positive-definite assumption for ൌ ׏ ݂ሺ࢞ ² If ׏ ݂ሺ࢞ is positive-definite, then a factorization of the form: ׏ ݂ሺ࢞ ሻ ൌ ࡸࡰࡸ ³ where ݀ ௜௜ ൐ Ͳ ³ can be used to solve for the resulting system of linear equations: ሺࡸࡰࡸ ሻࢊ ൌ െ׏݂ሺ࢞ ² If at any point D is found to have negative entries, i.e., if ݀ ௜௜ ൑ Ͳ ³ then it should be replaced by a positive value, such as ȁ݀ ௜௜ ȁ ² This correction amounts to adding a diagonal matrix E , such that ׏ ݂ሺ࢞ ሻ ൅ ࡱ is positive-definite. An algorithm for Newton’s method is given below. Get Help Now Go to for more info Need help with your dissertation? Get in-depth feedback & advice from experts in your topic area. Find out what you can do to improve the quality of your dissertation!
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 142 ±umerical Optimization Methods Newton’s Method (Griva, Nash, & Sofer, p. 373): Initialize: Choose ³ specify ߳ For ݇ ൌ Ͳǡͳǡ ǥ 1. Check convergence: If ԡ׏݂ሺ࢞ ሻԡ ൏ ߳ ³ stop 2. Factorize modified Hessian as ׏ ݂ሺ࢞ ሻ ൅ ࡱ ൌ ࡸࡰࡸ DQG VROYH ሺࡸࡰࡸ ሻࢊ ൌ െ׏݂ሺ࢞ IRU 3. Perform line search to determine ߙ and update the solution estimate as ௞ାଵ ൌ ࢞ ൅ ߙ Rate of Convergence. Newton’s method achieves quadratic rate of convergence in the close neighborhood of the optimal point, and superlinear rate of convergence otherwise. Moreover, due to its high computational and storage costs, classic Newton’s method is rarely used in practice. 7.3.4 Quasi-Newton Methods Quasi-Newton methods that use low-cost approximations to the Hessian matrix are the among most widely used methods for nonlinear problems. These methods represent a generalization of one- dimensional secant method, which approximates the second derivative as: ݂ ᇱᇱ ሺݔ ሻ ؆ ሺ௫ ሻି௙ ሺ௫ ೖషభ ି௫ ೖషభ ² In the multi-dimensional case, the secant method translates into the following: ׏ ݂ሺ࢞ ሻሺ࢞ െ ࢞ ௞ିଵ ሻ ؆ ߘ݂ሺ࢞ ሻ െ ߘ݂ሺ࢞ ௞ିଵ (7.25) Thus, if the Hessian is approximated by a positive-definite matrix H k , then H k then is required to satisfy the following secant condition: ሺ࢞ െ ࢞ ௞ିଵ ሻ ൌ ߘ݂ሺ࢞ ሻ െ ߘ݂ሺ࢞ ௞ିଵ (7.26) Whereas, the above condition places ݊ constraints on the structure of H k , further constraints may be added to completely specify H k as well as to preserve its symmetry.
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