Lecture 1 - Introduction to Optimization

# Lecture 1 - Introduction to Optimization - What is...

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Lecture 1 Introduction to Optimization UCSD Math 171A: Numerical Optimization Philip E. Gill ( pgill@ucsd.edu ) Monday, January 5th, 2009 UCSD Center for Computational Mathematics Slide 1/42, Monday, January 5th, 2009 What is optimization? Webster’s dictionary: Optimization the process or method for making something (design, system, decision) as fully perfect, functional or eﬀective as possible .” UCSD Center for Computational Mathematics Slide 2/42, Monday, January 5th, 2009 How do we optimize something? Formulate a mathematical model of a given situation or resource for which “optimizing” means minimizing or maximizing a function: f ( x , y , z , . . . ) the ”objective function” subject to restrictions on the values that x , y , z , . . . , can take. For example, x may need to be nonnegative , i.e., x 0. UCSD Center for Computational Mathematics Slide 3/42, Monday, January 5th, 2009 The problems to be considered In general we have “constraint functions” a ( x , y , z , . . . ) 0 b ( x , y , z , . . . ) 0 . . . . . . For example, x and y may need to be outside a circle, i.e., x 2 + y 2 - 1 0 UCSD Center for Computational Mathematics Slide 4/42, Monday, January 5th, 2009

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Examples from calculus minimize x x 2 + 1 The objective function f ( x ) = x 2 + 1 is minimized at x * = 0 The derivative is zero at x * df dx = f 0 ( x ) = 2 x = 0 UCSD Center for Computational Mathematics Slide 5/42, Monday, January 5th, 2009 Examples from calculus Similarly, minimize x , y x 2 + y 2 + 1 The function f ( x ) = x 2 + y 2 + 1 is minimized at x * = 0, y * = 0. The partial derivatives are zero at x * and y * f x = 2 x = 0 f y = 2 y = 0 UCSD Center for Computational Mathematics Slide 6/42, Monday, January 5th, 2009 Some examples of optimization problems Air travel: Ticket payment An optimization algorithm checks your credit card transaction Flight attendants A ﬂight scheduling algorithm assigned the ﬂight crew to your plane Aircraft design An optimization algorithm was used to design shape of the plane Passengers NBA game locations and dates are determined using optimization UCSD Center for Computational Mathematics Slide 7/42, Monday, January 5th, 2009 Design technology: Yacht design Optimal design of America’s Cup boats Space Shuttle The take-oﬀ and re-entry trajectories calculated to minimize structural stress and heating Determination of the surface temperature using sensors Minimization of fuel consumption and maximization of payload Calculation of trajectories for spacecraft Optimal control of long-term interplanetary missions Design of orbiters UCSD Center for Computational Mathematics Slide 8/42, Monday, January 5th, 2009
Inverse problems: Medical imaging Geophysical applications Estimating the Earth’s electrical conductivity

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## This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Lecture 1 - Introduction to Optimization - What is...

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