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ch22

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351 Part F. Optimization. Graphs CHAPTER 22 Unconstrained Optimization. Linear Programming SECTION 22.1. Basic Concepts. Unconstrained Optimization, page 936 Purpose. To explain the concepts needed throughout this chapter. To discuss Cauchy’s method of steepest descent or gradient method, a popular method of unconstrained optimization. Main Content, Important Concepts Objective function Control variables Constraints, unconstrained optimization Cauchy’s method SOLUTIONS TO PROBLEM SET 22.1, page 939 2. The line of approach is tangent to a particular curve C: ƒ( x ) const, the point of contact P giving the minimum, whereas the next gradient of C at P is perpendicular to C. 4. ƒ( x ) ( x 1 0.5) 2 2( x 2 1.5) 2 4.75. The computation gives: Step x 1 x 2 ƒ( x ) 1 0.25342 1.5206 4.6884 2 0.49351 1.4805 4.7492 3 0.49680 1.5003 4.7500 6. ƒ( x ) ( x 1 4) 2 0.1( x 2 5) 2 4. The computation gives: Step x 1 x 2 ƒ( x ) 1 4.0240 1.4016 5.2958 2 3.7933 4.8622 4.0449 3 4.0008 4.8760 4.0015 4 3.9929 4.9952 4.0000 5 4.0000 4.9957 4.0000 6 3.9997 5.0000 4.0000 7 4.0000 5.0000 4.0000 8. The calculation gives for Steps 1 5: x 1 x 2 ƒ( x ) 1.33333 2.66667 5.3334 3.55556 1.77778 9.4815 2.37037 4.74074 16.8560 6.32099 3.16049 29.9662 4.21399 8.42798 53.2731 im22.qxd 9/21/05 1:53 PM Page 351

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352 Instructor’s Manual This is the beginning of a broken line of segments spiraling away from the origin. At the corner points, ƒ is alternatingly positive and negative and increases monotone in absolute value. 10. ƒ( x ) x 1 2 x 2 gives z ( t ) x t [ 2 x 1 , 1 ] [ (1 2 t ) x 1 , x 2 t ] , hence g ( t ) (1 2 t ) 2 x 1 2 x 2 t , g ( t ) 4(1 2 t ) x 1 2 1 0. From this, 1 2 t , t . For this t , z ( t ) [ , x 2 ] . From this, with x 1 1, x 2 1, we get successively z (1) [ 1 _ 4 , 1 1 _ 2 1 _ 8 ] T z (2) [ 1, 1 2 1 _ 2 1 _ 8 2 ] T z (3) [ 1 _ 4 , 1 3 1 _ 2 2 1 _ 8 2 ] T etc. The student should sketch this, to see that it is reasonable. The process continues indefinitely, as had to be expected. 12. CAS Experiment. (c) For ƒ( x ) x 1 2 x 2 4 the values converge relatively rapidly to [0 0] T , and similarly for ƒ( x ) x 1 4 x 2 4 . SECTION 22.2. Linear Programming, page 939 Purpose. To discuss the basic ideas of linear programming in terms of very simple examples involving two variables, so that the situation can be handled graphically and the solution can be found geometrically. To prepare conceptually for the case of three or more variables x 1 , • • • , x n . Main Content, Important Concepts Linear programming problem Its normal form. Slack variables Feasible solution, basic feasible solution Optimal solution Comments on Content Whereas the function to be maximized (or minimized) by Cauchy’s method was arbitrary (differentiable), but we had no constraints, we now simply have a linear objective function, but constraints, so that calculus no longer helps. No systematic method of solution is discussed in this section; these follow in the next sections. 1 8 x 1 2 1 2 1 4 x 1 1 8 x 1 2 1 2 1 4 x 1 2 im22.qxd 9/21/05 1:53 PM Page 352
Instructor’s Manual 353 SOLUTIONS TO PROBLEM SET 22.2, page 943 2. No. For instance, ƒ 5 x 1 2 x 2 gives maximum profit ƒ 12 for every point on the segment AB because AB has the same slope as ƒ const does.

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ch22 - im22.qxd 1:53 PM Page 351 Part F Optimization Graphs...

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