IEOR160 midterm - IEOR 160 Fall 2010 Md term exam October...

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Unformatted text preview: IEOR 160 Fall 2010 Md term exam. October 20 Answer all questions. Stan each answer on a new page. The point value of each question is shown after the number 1 . (40) f(x) = (x-a)2 , where x is a scalar. Constraintsc’ b <= x <= c; In each of the following cases: Write the Kuhn-Tucker conditions, and solve if possible for (i) optimal value of x (ii) the optimal objective value and (iii) the rate of change of the optimal objective with respect the parameters ‘ and c. Explain your work. . ”Q A A. Minimize f(x) and explain if the solution is a global minimum for each of the following conditions: 1. a<b;' 2. b<a<c 3. a>c B. Maximize f(x) and explain if the solution is a global maximum for each of the following conditions: 1. a < b; ' I 2. b < a < c 3. a > c 2. (5) Problem: minimize f(x) a differentiable function; x is a vector in 11 space. ~ Themethedef—steepest—descent WWW»W«edifihe-dkeeéefiefisemhforabm—< solution, and t is the distance moved in that direction. The norm of d = 1; How is the vector (1 calculated? 3. (1-5) The set S : the set of vectors x such that g (x) <= bj for j = 1..n Prove : if all the fimctions gj(x) are convex than S is convex. 4. (20) Solve and explain2 your solution procedure maximizez=-—x12+8x,-xz2 +12x2 s.t. X” X; <=8 0 <= X1 <= 5 0 <= X2 <= 4 Is the solution a global maximum? Explain What is the rate of change\of the optimal objective with respect to the constants in the constraints? 5. You have entered a race that involves running and swimming. The starting position it is on land at (xl,yl); x1 is < 0; The finish point f is a small island in a lake at coordinates (x2,y2) where x2 >0 The lake shore is the straight line y= 0 Your running speed "- uur swimming speed @hr. Where should you enter the water? ...
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