hw1 - 2010

# hw1 - 2010 - Homework No. 1 Due date: October 8, 2010 1....

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Homework No. 1 Due date: October 8, 2010 1. Plot the contours (in the feasible region only) of a 2-dimensional function z x x (, ) 12 subject to 0 1 ≤≤ x a , 0 2 x b under the assumption that the function is: (a) Strictly convex (b) Convex with multiple minima (c) Linear (d) Constrained by x x c + = (in addition to the previous two constraints) 2. Show that the sum of two convex functions is a convex function. 3. Use the Kuhn-Tucker conditions to solve the following problem min ( , ) ( ) ( ) zx x x x 1 2 2 2 42 34 = + subject to x x 5 +≥ x x 1 2 1 2 Show that the solution is unique. 4. Use Lagrangian to solve the program min ( , ) x xx x 1 2 2 2 53 27 = + + subject to 25 x x + = x x 1 2 0 0 Plot the feasible region. 5. Use the golden section to find, within ± 0.15 of the optimum, the value of x that minimizes the function zx x x x () m a x( ) , ( ), =− R S T U V W 2 3 3 5 2 such that 08 x 6. Use the bisection method to find the minimum of x x () = + 1 2 over the interval [0, 2]. Determine the optimal value of x within 5% of the initial interval.

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## This note was uploaded on 12/24/2011 for the course CIVL 561 taught by Professor Hklo during the Fall '10 term at HKUST.

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hw1 - 2010 - Homework No. 1 Due date: October 8, 2010 1....

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