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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 #10: TUE, OCT 2, 2007 PRINT DATE: AUGUST 21, 2007 (GRAPHICAL4) Contents 2. Graphical Overview of Optimization Theory (cont) 1 2.9. Constrained Optimization: Several Variables (cont) 1 2.10. The Bizarre Exception 10 3. Envelope Theorem and Implicit Function Theorem 11 2. Graphical Overview of Optimization Theory (cont) 2.9. Constrained Optimization: Several Variables (cont) We are now going to look at a number of special cases of the general problem of maximizing an objective function subject to constraint(s). (1) A single linear inequality constraint (Fig. 24): • The constraint is the condition that g ( x ) ≤ b . That is, the only x ’s we are allowed to consider are x ’s for which the function g assigns values weakly less than b . • Three candidates in the picture a , b and c . Each curve represents a level set of some function thru one of these points. Note: each level set belongs to a different function! • Point b isn’t a maximum; upper contour set intersects with constraint set; Point c isn’t either. Point a is the only one with the property that there’s nothing in the strict upper contour set that is also in the constraint set. 1 2 LECTURE #10: TUE, OCT 2, 2007 PRINT DATE: AUGUST 21, 2007 (GRAPHICAL4) in same point direction arrows constraint arrow objective arrow Lower contour set of g corresponding to b y x 1 x 2 g ( x ) = b a b c Figure 24. Constrained max problem with one inequality constraint • Now put in the arrows. Call the first the objective arrow (i.e., the gradient of the objec tive function) and the second the constraint arrow (i.e., the gradient of the constraint function). Which way do the objective arrows point? – arrow points in direction of increase – this must be the upper contour set – upper contour set is convex – Conclude: we know that the arrow for a level set points into the corresponding upper contour set, which for a quasiconcave function is the convex set bounded by the set. • Which way does the constraint arrow point? Must be NE because the lower contour set is SW. • Now characterize the condition for a maximum in terms of the arrows. Answer is that arrows lie on top of each other and point in the same direction (as opposed to lying on top of each other but point at 180 degrees to each other). • Conclude in terms of the mantra: arrow lies in the positive cone defined by the unique binding constraint. Question: Suppose that for each of the level sets of f , the gradient vector at every point on this level set points SW as in point c ? What can you conclude about the solution to the maximization problem? ARE211, Fall 2007 3 y x 1 x 2 a b c Figure 25. Constrained max problem with one linear equality constraint Answer: There is no solution. Certainly there can’t be a solution in which the constraint is binding. And there can’t be an unconstrained solution either. You canis binding....
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 Fall '07
 Simon
 objective function, Constraint, Gradient, Level set, nonnegative cone

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