ch.6 - Chapter 6 Optimization Method of Lagrange...

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Chapter 6. Optimization: Method of Lagrange Multipliers 6.1. Constrained Optimization In Chapter 4 we have studied a method of searching and classifying all stationary points as local extrema and saddle points. In this chapter we shall consider the problem of maximizing or minimizing a function of n variables subject to the constraint that the points ( x 1 , . . . , x n ) T must satisfy certain equations. Here we illustrate a systematic way to deal with constrained optimization problems. The method of Lagrange multipliers replaces finding stationary points of a constrained func- tion f : R n -→ R of n variables with k constraints to finding stationary points of an uncon- strained function L in n + k variables. This method introduces a new variable, known as the Lagrange multiplier , for each constraint and defines a new function, called the Lagrangian in- volving the original function, the constraints and the Lagrange multipliers. The k constraints are often given by a vector-valued function g : R n -→ R k of n variables: g ( x 1 , . . . , x n ) = ( g 1 ( x 1 , . . . , x n ) , . . . , g k ( x 1 , . . . , x n )) T . Let us begin with the simplest case. Let f : R 2 -→ R and g : R 2 -→ R be both real-valued functions of two variables. Consider the problem of maximizing/minimizing f ( x, y ) subject to the single constraint g ( x, y ) = 0 . We sometimes write max f ( x, y ) ( or min f ( x, y ) ) subject to g ( x, y ) = 0 . Consider the contour curves f ( x, y ) = c of f for various values of c , and the contour curve g ( x, y ) = 0 of g that represents the constraint of our problem. Now fix our attention on the contour curve g ( x, y ) = 0 . In general this contour curve will intersect with different contour curves f ( x, y ) = c , which means the value of f can change whilst moving along the contour curve g ( x, y ) = 0 : When the contour curve g ( x, y ) = 0 meets the contour f ( x, y ) = c 0 for some c 0 R tangen- tially at the point ( a, b ) T , the value of f will not be increased or decreased, that is, the directional derivative of f along the tangential direction of the curve g ( x, y ) = 0 at ( a, b ) T is equal to 0 . Recall that the direction in which f has zero change is given by the direction orthogonal to f . In other words, the gradient vector f must be normal to the curve g ( x, y ) = 0 at ( a, b ) T . On the other hand, the gradient vector g is of course normal to the contour g ( x, y ) = 0 at ( a, b ) T . Geometrically, what we have deduced is that the contour curves f ( x, y ) = c 0 and g ( x, y ) = 0 have a common tangent at ( a, b ) T , and the gradient vectors f ( a, b ) and g ( a, b ) must point in the same or opposite directions: The point ( a, b ) T is sometimes called the constrained stationary point of f . Now we can write f ( a, b ) + λ g ( a, b ) = 0 for some scalar λ R , or equivalently ∂f ∂x ( a, b ) + λ ∂g ∂x ( a, b ) = 0 and ∂f ∂y ( a, b ) + λ ∂g ∂y ( a, b ) = 0 .
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