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MATH 417/517 - Linear Optimzation Homework Set No. 2
发布时间:2022-10-07
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MATH 417/517 - Linear Optimzation
Homework Set No. 2
3.1 (LP standard form) Bring the linear program
max x1 + 2x2 + 3x3 s.t. x1 + x3 > 0,
x1 - x3 < 0,
x1 + x2 = 1,
x1 , x2 > 0.
to standard form
min cT x s.t. Ax = b, x > 0,
with A e Rm ×n , b e Rm and c e Rn and write down A, b, c and n, m e N explicitly.
(3 P.)
3.2 (LP modelling) A hauling company has trucks (of the same size) a the locations A and B, namely 18 at location A and 12 at location B. These are to be sent to the terminals R,S and T where there are 11, 10 and 9 trucks needed, respectively. The distances (in km) from the locations to the terminals are given in the following table:
|
Terminal R |
Terminal S |
Terminal T |
Location A |
5 |
4 |
9 |
Location B |
7 |
8 |
10 |
The trucks are to be routed such that the kilometers driven are minimal and such
that the needs at every terminal are met.
a) Formulate this problem as a linear program.
b) Use linear algebra to reduce it to a two-dimensional problem in order to solve it graphically.
(3+8 P.)
3.3 (The full rank assumption on A) Let P = P (A, b) be a nonempty polyhedron
in standard form with
\- a1(T) -)
(- am(T) -)
such that rank A = k < m and ai1 , . . . , aik are linearly independent. Show that for (the polyhedron in standard form)
Q := {x e Rn ' ai}(T)x = bi} (j = 1, . . . , k), x > 0 }
we have Q = P .
(4 P.)
3.4 (Finitely many basic feasible points) Let b e Rm , A e Rm ×n with rank A = m and let P = P (A, b) be the polyhedron in standard form defined by (A, b). Show that P has at most finitely many basic feasible points.
(4 P.)
4.1 For v e Rn compute inf {vT x | x > 0 } . (2 P.)
4.2 (Dual of the dual is the primal problem) For (A, b, c) e Rm ×n × Rm × Rn consider the primal linear program
min cT x s.t. Ax = b, x > 0 (P)
and its dual
max bT y s.t. ATy < c. (D)
Show that the dual of the dual program is (equivalent to) the primal.
(4 P.)
4.3 Consider the primal linear program (P) and its dual (D) and show:
a) If (P) is unbounded then (D) is infeasible.
b) If (D) is unbounded then (P) is infeasible.
Afterwards fill in the table below.
|
inf(P) = -o |
inf(P) e R |
inf(P) = +o |
sup(D) = -o |
|
|
|
sup(D) e R |
|
|
|
sup(D) = +o |
|
|
|
Use the symbol ’文’ if the case for the tuple (inf(P), sup(D)) can occur, and ’×’ if this case cannot occur. Verify the positive case by an example, and cite a result from the lecture that explains the negative case.
(2+9 P.)
*4.4 (Equivalence of Farkas Lemma and strong duality) Show that the Farkas
Lemma (Proposition 2.4.5) follows from the strong duality theorem (Theorem 2.5.6).
(4 P.)