Lecture-5-1-Optimization-slices

Lecture-5-1-Optimization-slices - BME 5020 Computer and...

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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Optimization BME 5020 Computer and Mathematical Application in Bioengineering © 2006 Jingwen Hu October 3, 2006 • 1-D Unconstrained • 2+-D Unconstrained • Constrained Contents
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Introduction • Root finding and optimization are related, both involve guessing and searching for a point on a function. • Fundamental difference is: • Root finding is searching for zeros of a function or functions • Optimization is finding the minimum or the maximum of a function of several variables.
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Introduction
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background •A n optimization or mathematical programming problem generally be stated as: Find x , which minimizes or maximizes f(x) subject to Where x is an n -dimensional design vector , f(x) is the objective function , d i (x) are inequality constraints , e i (x) are equality constraints , and a i and b i are constants * , , 2 , 1 ) ( * , , 2 , 1 ) ( p i b x e m i a x d i i i i K K = = =
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Mathematical Background • Optimization problems can be classified on the basis of the form of f(x): • If f(x) and the constraints are linear, we have linear programming. • If f(x) is quadratic and the constraints are linear, we have quadratic programming. • If f(x) is not linear or quadratic and/or the constraints are nonlinear, we have nonlinear programming. • When equations(*) are included, we have a constrained optimization problem; otherwise, it is unconstrained optimization problem.
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University 1-D Unconstrained Optimization •I n multimodal functions, both local and global optima can occur. In almost all cases, we are interested in finding the absolute highest or lowest value of a function.
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University How do we distinguish global optimum from local one? • By graphing to gain insight into the behavior of the function. • Using randomly generated starting guesses and picking the largest of the optima as global. • Perturbing the starting point to see if the routine returns a better point or the same local minimum.
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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Golden-Section Search •A unimodal function has a single maximum or a minimum in the a given interval. For a unimodal function: • First pick two points that will bracket your extremum [ x l , x u ]. • Pick an additional third point within this interval to determine
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This note was uploaded on 04/04/2008 for the course BME 5020 taught by Professor King during the Fall '06 term at Wayne State University.

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Lecture-5-1-Optimization-slices - BME 5020 Computer and...

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