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How does Microsoft Excel solver deal with nonlinear problems – A simple example

How does Microsoft Excel solver deal with nonlinear problems – A simple example. 劉亮志. Recall our question. The equation for GARCH(1,1) is where Set , Then we get. Later, we met an example about MLE method :

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How does Microsoft Excel solver deal with nonlinear problems – A simple example

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  1. How does Microsoft Excel solver deal with nonlinear problems – A simple example 劉亮志

  2. Recall our question

  3. The equation for GARCH(1,1) is where Set , Then we get

  4. Later, we met an example about MLE method : Consider the problem of estimating the variance of variable Xfrom m observations on X when underlying distribution is normal with zero mean. Assume that the observations are . Denote the variance v . The likelihood of being observed is defined as the probability density function for X when . This is Then the joint density function of this m observations is (**)

  5. Using MLE, the best estimate of v is the value that maximizes the expression (**). Maximizing (**) is same as maximizing Let Then we want to maximize the expression

  6. Wewonder what’s the value of that maximizes subject to the boundary conditions

  7. How does Microsoft Excel solver deal with nonlinear programs? (1) the long …long way to find the method out (2) basic idea and a simple example (3) reference

  8. Long way Microsoft Office Online • Excel adopts the generalized reduced gradient (GCG) • methodto deal withnonlinear problems. • GCG originates from the method of reduced gradient • of Wolf. • More specifically, the question (MLE) we met is a nonlinear • programming problem with linear constraints. http://office.microsoft.com/zh-tw/

  9. Basic idea of method of reduced gradient

  10. Summary of Reduced Gradient Algorithm

  11. A simple example

  12. Iterative 1

  13. Iterative 2

  14. Iterative 3

  15. Reference

  16. Mokhtar s. Bazraa, Hanif D. Sherali ,and C. M. Shetty. Nonlinear Programming: Theory and Algorithm second edition Section 10.6 • Wenyu Sun, and Ya-Xiang Yuan. Optimization Theory and Methods: Nonlinear Programming Section 11.3 • Hong-Tau Lee, Sheu-Hua Chen, He-Yau Kang. A Study of Generalized Reduced Gradient Method with Different Search Directions • Daniel Fylstra, Leon Lasdon, John Waston, Allen Waren. Design and Use of the Microsoft Excel Solver.

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