IEx86-StudySet-1

(Dantzig-Wolfe IP Block-diagonal Matrix) For the Questions 3 and 4, consider the following optimization problems. Identify a bordered block- diagonal structure in the constraint matrix of this problem by visual inspection. Then, solve this optimization problem via Dantzig-Wolfe reformulation using the block-diagonal structure identified above. Write down clearly your master problem and subproblems. No need to do all the iterations, you can stop after three iterations. Check if the solution you found is integral (i.e., you can termi- nate the DW formulation at the root node) or fraction (i.e., you need to branch further). Q3) maximize 7 x 1 + 4 x 2 + 5 x 3 + 2 x 4 + 2 x 5 + 9 x 6 subject to x 1 + 2 x 2 + x 3 + x 4 + 2 x 5 + 3 x 6 ≤ 20 , x 1 + x 3 ≤ 6 , 2 x 2 + x 5 ≤ 5 , x 4 + 4 x 6 ≤ 9 , 2 x 1 + 3 x 3 ≤ 12 , 3 x 2 + 2 x 5 ≤ 12 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ≥ 0 and integer. ( P 3 ) Q4) maximize 2 x 1 + 4 x 2 + 3 x 3 + x 4 subject to x 1 + x 2 + x 3 + x 4 ≤ 7 , x 1 + x 3 ≤ 5 , x 1 + 2 x 4 ≤ 6 , x 1 + 3 x 4 ≤ 7 , 2 x 2 + 3 x 3 ≤ 11 , x 2 + x 3 ≤ 5 , x 1 , x 2 , x 3 , x 4 ≥ 0 and integer. ( P 4 ) Q5) (Lagrangian Relaxation) Consider the following optimization prob- lem: minimize 12 x 1 + 9 x 2 + 10 x 3 + 14 x 4 subject to 8 x 1 + 5 x 2 + 6 x 3 + 9 x 4 ≥ 14 , x 1 + x 3 ≤ 1 , x 1 + x 4 ≤ 1 , x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } . ( P 5 ) Apply Lagrangian relaxation by relaxing each of the constraints separately. Re- port the Lagrangian bound you found for each case. Use both Cutting Plane Method and Subgradient optimization in each relaxation. 2