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Lecture11 - Engineering Analysis ENG 3420 Fall 2009 Dan C...

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Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00
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2 Lecture 11 Lecture 11 Last time: Newton-Raphson The secant method Today: Optimization Golden ratio makes one-dimensional optimization efficient. Parabolic interpolation locate the optimum of a single-variable function. fminbnd function determine the minimum of a one-dimensional function. fminsearch function determine the minimum of a multidimensional function . Next Time Linear algebra
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Optimization Critical for solving engineering and scientific problems. One-dimensional versus multi-dimensional optimization. Global versus local optima. A maximization problem can be solved with a minimizing algorithm. Optimization is a hard problem when the search space for the optimal solution is very large. Heuristics such as simulated annealing, genetic algorithms, neural networks. Algorithms Golden ratio makes one-dimensional optimization efficient. Parabolic interpolation locate the optimum of a single-variable function. fminbnd function determine the minimum of a one-dimensional function. fminsearch function determine the minimum of a multidimensional function. How to develop contours and surface plots to visualize two- dimensional functions. 3
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Optimization Find the most effective solution to a problem subject to a certain criteria. Find the maxima and/or minima of a function of one or more variables.
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One- versus multi-dimensional optimization One-dimensional problems involve functions that depend on a single dependent variable -for example, f ( x ). Multidimensional problems involve functions that depend on two or more dependent variables - for example, f ( x , y )
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Global versus local optimization G lobal optimum the very best solution. L ocal optimum solution better than its immediate neighbors. Cases that include local optima are called multimodal . Generally we wish to find the global optimum.
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One- versus Multi-dimensional Optimization One-dimensional problems involve functions that depend on a single dependent variable -for example, f ( x ). Multidimensional problems involve functions that depend on two or more dependent variables - for example, f ( x , y )
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Euclid’s golden number Given a segment of length the golden number is determined from the condition: The solution of the last equation is 8 2 1 y y + 2 1 y y = ϕ 0 1 2 2 2 1 2 2 1 = - - + = ϕ ϕ y y y y y 680133 . 1 2 5 1 = + = ϕ
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