HW8 - ε = 0 , by construct-ing a dual optimal point z *...

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ELE 704 Optimization HW8 Due 1 May, 2007 1. Show that the weak max-min inequality sup z Z inf w W f ( w , z ) inf w W sup z Z f ( w , z ) always holds, with no assumptions on f : R n × R m R , W R n and Z R m 2. Consider the following optimization problem min x 2 1 + x 2 2 s.t. ( x 1 - 1) 2 + ( x 2 - 1) 2 1 ( x 1 - 1) 2 + ( x 2 + 1) 2 1 where x R 2 . (a) Plot the feasible set and the level sets of the objectives. Find the optimal point x * and the optimal value p * . (b) Give the KKT conditions. Do there exist multipliers λ * 1 and λ * 2 that prove that x * is optimal? (c) Derive and solve the Lagrange dual problem. Does strong duality hold? 3. Consider the pair of primal and dual linear programming problems min ( c + ε d ) T x s.t. Ax ± b + ε f and max - ( b + ε f ) T z s.t. A T z + c + ε d = 0 z ² 0 where A = - 4 12 - 2 1 - 17 12 7 11 1 0 - 6 1 3 3 22 - 1 - 11 2 - 1 - 8 , b = 8 13 - 4 27 - 18 , f = 6 15 - 13 48 8 , c = 49 - 34 - 50 - 5 , d = 3 8 21 25 and ε is a parameter. 1
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(a) Prove that x * = ± 1 1 1 1 ² T is optimal when
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Unformatted text preview: ε = 0 , by construct-ing a dual optimal point z * that has the same objective value as x * . Are there any other primal or dual optimal solutions? (b) Give and explicit expression for the optimal value p * ( ε ) as a function of ε on an interval that contains ε = 0 . Specify the interval on which your expression is valid. Also give explicit expressions for the primal solution x * ( ε ) and the dual solution z * ( ε ) as a function of ε, on the same interval. (Hint: First calculate x * ( ε ) and z * ( ε ) , assuming that the primal and dual constraints that are active at the optimum for ε = 0 , remain active at the optimum for values of ε around 0. Then verify that this assumption is correct.) 2...
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HW8 - ε = 0 , by construct-ing a dual optimal point z *...

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