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notes_12 - Lecture Notes For Optimization Theory &...

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Lecture Notes For Optimization Theory &Ana lys is (ESE 304) Michael A. Carchidi November 24, 2010 Chapter12-Non l inearProgramm ing The following notes are based on the text entitled: Introduction to Mathemat- ical Programming by Wayne L. Winston and Munirpallam Venkataramanan (4th edition), and these can be viewed at https://courseweb.library.upenn.edu/ after you log in using your PennKey user name and Password. 12.1 A Brief Review Of Di f erential Calculus So far, we have been concerned with the Linear Programming (LP) problem in which we seek to optimize a linear objective function subject to linear constraints. In this chapter we consider the Nonlinear Programming (NLP) problem in which either the objective function is nonlinear or one or more of the constraints are nonlinear. The principle methods used to study NLP problems are from the Di f erential Calculus and so we begin our study of NLP by f rst reviewing some key facts in Di f erential Calculus. We do assume however, that the student has already taken a course in Di f erential Calculus and is familiar with its techniques. Functions of a Single Variable A function of a single variable x is a rule which assigns to each real number in some set A (called the domain of the function) a unique real number in some set B (called the range of the function). We represent this by the notation f : A B ,
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and for a given real number x in A ,weass igntherea lnumber y = f ( x ) (read ” f of x ”) in B .No tetha t A and B are subsets of the set of all real numbers R .W e represent this symbolically by writing x input f y = f ( x ) output and the input to the function ( x ) is called its argument . For example, suppose that f : A B is de f ned so that y = f ( x )= 1 x 2 +1 , then A = R and B = { y | 0 <y } , or suppose that f : A B is de f ned so that y = f ( x 2 x x 2 , then A = R and B = { y | 1 y 1 } . As a third example, suppose that f : A B is de f ned so that y = f ( x 1 1+ x , then A = { x | 0 x } and B = { y | 0 1 } . A very useful way to ”picture” a function of one variable is through its graph, which is a plot of y = f ( x ) versus x in an xy plane. The following three graphs show the three functions considered above. x 4 2 0 -2 -4 1 0.8 0.6 0.4 0.2 Plot of y =1 / ( x 2 +1) x 2
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x 4 2 0 -2 -4 1 0.5 0 -0.5 -1 Plot of y =2 x/ ( x 2 +1) versus x x 5 4 3 2 1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Plot of y =1 / (1 + x ) x Limits The limit concept is at the base of the di f erential calculus. Suppose that f : A B is a function, the equation lim x a f ( x )= c means that as x gets arbitrary close to a (but not equal to a ), the value of f ( x ) gets arbitrary close to c .N o t eth a t a need not be in the domain of the function f . For example, if f : A B such that f ( x e x 1 x 3
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then A = { x | x 6 =0 } and lim x 0 f ( x )=1 . Note that it is possible that the above limit does not exist. For example if f : A B such that f ( x )= 1 x then A = { x | x 6 } and lim x 0 f ( x ) does not exist.
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notes_12 - Lecture Notes For Optimization Theory &amp;...

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