Prove that the dual of (P1) has a finite optimal valı
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.6: Quadratic Functions
Problem 30E
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Question
![3 Duality
(3.1) Consider the following polyhedron in standard form:
P = {x € R" : Ax = b,
where A e Rmxn and b € Rm are given and x € Rn.
(i) Consider the Phase 1 problem for solving (P):
(P1) min{ea : Ax + a =b, x≥ 0, a ≥ 0},
where a € Rm and e E Rm denotes the vector of all ones, i.e., e = [1,1, ..., 1]T. Write
down the dual of (P1), explaining clearly each step in your solution.
(ii) Prove that the dual of (P1) has a finite optimal value.
(3.2) Determine the set of all optimal solutions for the following linear program using the graphical
method:
(P)
min
s.t.
X1 + X2 +
X1
X2
+
X1 +
2x2
X1
x2
x ≥ 0},
Suppose that b ≥ 0.
"
9
x3
X3
2x3
x3
-
X4
X4
2x4
X4
=
=
≥ 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2F1b1f0040-0777-4749-a093-50886c141c5f%2Ffb8l8qr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3 Duality
(3.1) Consider the following polyhedron in standard form:
P = {x € R" : Ax = b,
where A e Rmxn and b € Rm are given and x € Rn.
(i) Consider the Phase 1 problem for solving (P):
(P1) min{ea : Ax + a =b, x≥ 0, a ≥ 0},
where a € Rm and e E Rm denotes the vector of all ones, i.e., e = [1,1, ..., 1]T. Write
down the dual of (P1), explaining clearly each step in your solution.
(ii) Prove that the dual of (P1) has a finite optimal value.
(3.2) Determine the set of all optimal solutions for the following linear program using the graphical
method:
(P)
min
s.t.
X1 + X2 +
X1
X2
+
X1 +
2x2
X1
x2
x ≥ 0},
Suppose that b ≥ 0.
"
9
x3
X3
2x3
x3
-
X4
X4
2x4
X4
=
=
≥ 0.
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