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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:
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(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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 Spring '10
 TanBanPin
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