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hmk_6_s - ESE304 Introduction to Optimization(Homework#6...

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ESE304 - Introduction to Optimization (Homework #6) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (20 points) A river runs along the curve described in the xy plane by the equation y = 2 x 2 + 19 . A fi sherman is driving along a straight road described in the xy plane by the equation y = 4 x . After parking this car somewhere along the road, the fi sherman must walk in a straight line to the river. The fi sherman would like to park his car along the road so that he has to walk the least amount of distance to the river. At what point ( x 1 , y 1 ) should he park his car and at what point ( x 2 , y 2 ) will he do his fi shing? The River (Curve) and The Road (Straight) a.) (8 points) First you want to formulate the problem which will minimize the distance he must walk. Use the four decision variables x 1 , y 1 , x 2 and y 2 . b.) (4 points) Set up the solution to the problem using the method of Lagrange multipliers. You should end up with six (6) equations involving six (6) unknowns. c.) (8 points) Solve the problem using the method of Lagrange multipliers. Note that in order to simplify the algebra, you may seek to minimize the square of the distance walked rather than the distance walked. ––––––––––––––––––––––––––––––––––––
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–––––––––––––––––––––––––––––––––––– Problem #2 (20 points) Determine the maximum and minimum values of the linear function f ( x 1 , x 2 ) = x 1 + 4 x 2 over the feasible region R = { ( x 1 , x 2 ) | 4 x 1 x 2 , x 1 + x 2 2 17 , 0 x 1 , 0 x 2 } . –––––––––––––––––––––––––––––––––––– Problem #3 (20 points) Consider the nonlinear function f ( x 1 , x 2 ) = 2 x 3 1 + 3 x 2 1 x 2 + 3 x 1 x 2 2 + x 3 2 for all points ( x 1 , x 2 ). Determine the range of values for x 1 and x 2 so that this function is ( a ) convex ( b ) concave ( c ) neither convex nor concave. –––––––––––––––––––––––––––––––––––– Problem #4 (20 points) Determine the (a) maximum and (b) minimum values for z = 3 x 1 +4 x 2 over the feasible region R = n ( x 1 , x 2 ) | ( x 1 1) 2 x 2 and x 2 1 + x 2 2 1 o . –––––––––––––––––––––––––––––––––––– Problem #5 (20 points) Use graphical methods to determine the dimensions of a right triangle that has the largest possible area, given that the perimeter cannot be larger than P . Your fi nal answer should be in terms of P . –––––––––––––––––––––––––––––––––––– 2
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–––––––––––––––––––––––––––––––––––– Solution to Problem #1 a.) It is clear that we seek to minimize the distance D ( x 1 , y 1
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