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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #9: THU, SEP 27, 2007 PRINT DATE: AUGUST 21, 2007 (GRAPHICAL3) Contents 2. Graphical Overview of Optimization Theory (cont) 1 2.6. Level Sets, upper and lower contour sets and Gradient vectors (cont) 1 2.7. Quasiconcavity, quasiconvexity 3 2.8. Strict Quasiconcavity 4 2.9. Constrained Optimization: Several Variables 9 2. Graphical Overview of Optimization Theory (cont) 2.6. Level Sets, upper and lower contour sets and Gradient vectors (cont) Vectors : Recall that a vector in R n is a collection of n scalars. A vector in R 2 is often depicted as an arrow. Properly the base of the arrow should be at the origin, but often you see vectors that have been “picked up” and placed elsewhere. Example below. Gradient vectors : When economists draw level sets through a point, they frequently attach arrows to the level sets. These arrows are pictorial representation of the gradient vector , i.e., the slope of f at x , f prime ( x ). Its components are the partial derivatives of the function f , evaluated at x , i.e., ( f 1 ( x ) , ··· ,f n ( x )) 1 2 LECTURE #9: THU, SEP 27, 2007 PRINT DATE: AUGUST 21, 2007 (GRAPHICAL3) 5 10 5 10 5 10 15 20 25 30 35 40 45 50 Figure 18. Level set and gradient vector through a point Example : f ( x ) = 2 x 1 x 2 , evaluated at (2 , 1), i.e., f prime (2 , 1) = (2 x 2 , 2 x 1 ) = (2 , 4). Draw the level set through (2 , 1), draw the gradient through the origin, lift it up and place its base at (2 , 1). Generally, the gradient of a function with n arguments is a point in R n , and for this reason, you often see the gradient vector drawn in the domain of the function, e.g., for functions in R 2 , you often draw the gradient vector in the horizontal plane. The gradient vector points in the direction of steepest ascent : Consider Fig. 18. Let x denote the point in the domain where the first straight line touches the circle. The graph represents a nice symmetric mountain which you are currently about to scale. You are currently at the point x . You’re a macho kind of person and you want to go up the mountain in the steepest way possible. Ask yourself the question, looking at the figure. What direction from x is the steepest way up? Answer is: the direction perpendicular to the straight line. Draw an arrow pointing in this direction. Now the gradient vector of f at x is an arrow pointing in precisely the direction you’ve drawn. The following things about the gradient vector are useful to know: • its length is a measure of the steepness of the function at that point (i.e., the steeper the function, the longer is the arrow.) ARE211, Fall 2007 3 • as we’ve seen it is perpendicular to the level set at the point x • it points inside the upper contour set. Note Well: It could point into the upper contour set, but then go out the other side!...
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 Fall '07
 Simon
 Optimization, Mathematical analysis, Mathematical optimization, Convex function, NPP.

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