Tutorial 3 - NATIONAL UNIVERSITY OF SINGAPORE Department of...

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Unformatted text preview: NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 3 1. (a) For each of the following sets, decide whether it is representable as a polyhedron? (i) The set of all (x, y ) ∈ R2 satisfying the constraints x cos θ + y sin θ ≤ 1 ∀θ ∈ [0, π/2] x≥0 y≥0 . (ii) The set of all x ∈ R satisfying the constraint x2 − 8x + 15 ≤ 0. (b) Let f : Rn → R be a convex function and let a be some constant. Show that the level set S = {x ∈ Rn | f (x) ≤ a} is a convex set. 2. Consider the following constraints in a LP problem: 2x1 − x2 + 5x3 = −1 3x2 + x4 ≤ 5 7x1 − 4x3 + x4 ≥ 4 x1 , x2 , x4 ≥ 0. Identify all active constraints at each of the following points x = (x1 , x2 , x3 , x4 ) and determine whether it is a basic feasible solution: 5 (a) x = (2, 0, 2 , 0) , (b) x = (1, 3, 0, 1) , (c) x = (5, 1, −2, 2) , (d) x = (0, 0, − 1 , 5) . 5 3. Suppose the feasible set P in R2 is described by the following constraints: −2x1 3x1 −3x1 2x1 x1 + − − + 3 x2 2x2 2x2 3 x2 x2 ≥ 6 ≤ 6 ≤ −12 ≥ −3 ≥ 0 ≥ 0. (a) Sketch the feasible set P and find all extreme points of P . 1 (b) For each of the following objective, (i) min 2x1 + x2 (ii) max 4x1 − 6x2 (iii) max 3x1 + x2 compute the corresponding objective value at each extreme point of P found in part (a); and determine all optimal solutions if the LP is not unbounded. 4. Consider the polyhedron P = {(x1 , x2 , x3 ) ∈ R3 | x1 + x2 + x3 ≤ 2, x1 ≥ 0, x2 ≥ 1}. Does P contain a line? 5. Consider the following LP: max 4x1 + 3x2 − 2x3 s.t. 2x1 + x2 + 2x3 = 10 5x1 + 2x2 + 4x3 + x4 = 21 x1 , x 2 , x 3 , x 4 ≥ 0 (a) (i) Verify that columns A1 and A4 are linearly independent. (ii) Determine the inverse B−1 of the associated basis matrix B = (A1 , A4 ). Identify all basic variables and all nonbasic variables. (iii) Construct the associated basic solution. Is this solution feasible, i.e. xB ≥ 0? (b) Determine all basic feasible solutions of the LP problem and their corresponding objective values. 2 ...
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Tutorial 3 - NATIONAL UNIVERSITY OF SINGAPORE Department of...

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