201-Lect 4 - Background Week 4 Both root location and...

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1 Week 4 Week 4 Optimization (ch 13-16) Week 4 Background • Both root location and optimization involve guessing and searching for a point on a function • Root location involves searching for zeros of a function or functions, while optimization involves searching for either the minimum or the maximum Week 4 Background • Optimization deals with finding the “best result”, or optimum solution, of a problem • Engineers must continuously design devices and products that perform tasks in an efficient fashion. • Engineers are always confronting optimization problems that balance performances and limitations Week 4 Mathematical background • An optimization problem is generally stated as: Find x to minimize or maximize f( x ) subject to: d i ( x ) a i i=1,2,…,m e i ( x ) = b i i=1,2,…,p 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
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2 Week 4 1-D and multi-D optimization • One-dimensional problems involve functions that depend on a single dependent variable • Multi-dimensional involve functions that depend on two or more dependent variables Week 4 Optimization categories One-dimensional unconstrained optimization -Golden-section search - Quadratic interpolation - Newton’s method Multi-dimensional unconstrained optimization - Direct methods - Gradient methods Constrained optimization - linear programming Week 4 1-D unconstrained optimization • Local and global optima • Optimization in 1-D consists of bracketing and open methods Week 4 Golden-section search • The goal is to minimize function evaluations by replacing old values with new values, based on the following conditions: 618 . 0 2 1 5 1 2 1 2 0 1 2 1 0 = = = = + = l l R calculated is ratio golden the l l l l l l l
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3 Week 4 Golden-section algorithm • The golden-section method starts with two initial guesses, x l and x u , that bracket one local extremum of f ( x ). Next, two interior points x 1 and x 2 are chosen according to the golden ratio d x x d x x x x d u l l u = + = = 2 1 ) ( 2 1 5 Please note the x 2 is always to the left of x 1 x 1 x 2 Week 4
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This note was uploaded on 10/27/2009 for the course MECH 201 taught by Professor Weihuali during the Three '09 term at University of Wollongong, Australia.

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201-Lect 4 - Background Week 4 Both root location and...

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