MA4254: Discrete Optimization
Additional LP used material in Chapter 3
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/15
Duality
For A Rmn , the standard form L
MA4254: Discrete Optimization
Additional LP used material in Chapter 3
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2015
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/15
Duality
For A Rmn , the standard form L
MA4254: Discrete Optimization
Example on Branch and Bound
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/8
F1
LP Relax
Active list:
cfw_F1
Upper
bound:
min
s.t
MA4254: Discrete Optimization
Chapter 3: The Shortest Path
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/46
3.1 Dijkstra's Algorithm
In this section (section 3
MA4254: Discrete Optimization
Additional LP used material in Chapter 3
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2015
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/15
Duality
For A Rmn , the standard form L
MA4254: Discrete Optimization
Chapter 1: Introduction
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/61
Aims/Objectives:
Discrete optimization deals with proble
Simplex Method (Shortened version for MA4254)
A tableau implements the simplex method conveniently.
From now on, we'll use tableaus for computation and analysis.
A generic simplex tableau is as follows:
Basic
c
B
x
T
T 1 A
c cB B
x
B
1 A
Solution
cT B1 b
MA4254: Discrete Optimization
Chapter 4: The Greedy Algorithm and Computational
Complexity
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2015
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/42
4.1 Matroid
1935, m
MA4254: Discrete Optimization
Chapter 5: Algorithms for N PC
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/57
1. Exact methods: Branch and bound, cutting plane
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 4
Q1. Refer to the Primal-Dual Algorithm in the lecture notes.
(a) Show that every admissible column in the optimal basis of (RPJ ) rem
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 3
Solution or hint to selected questions
Q1. [Outline of proof: We start with the denition of extreme points. Suppose that x
is an extr
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 1
Solution to selected questions
Q1.
n
n
min
cij xij
i=1 j=1
n
xij = 1,
i = 1, , n
xij = 1,
s.t.
j = 1, , n
j=1
n
i=1
2
x cfw_0, 1n .
Q
MA4254: Discrete Optimization
Chapter 3: The Shortest Path
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/42
3.1 Dijkstra's Algorithm
In this section (section 3
MA4254: Discrete Optimization
Chapter 2: Total Unimodularity and its applications
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/20
2.1 Total Unimodularity: Den
MA4254: Discrete Optimization
Chapter 4: The Greedy Algorithm and Computational
Complexity
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/42
4.1 Matroid
1935, m
MA4254: Discrete Optimization
Chapter 3: The Shortest Path
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/46
3.1 Dijkstras Algorithm
In this section (section 3.
MA4254: Discrete Optimization
Chapter 1: Introduction
Pang Chin How Jerey
Department of Mathematics
National University of Singapore
August 2014
(Based on notes by Prof Sun Defeng)
NUS: MA4254: 1/61
Aims/Objectives:
Discrete optimization deals with proble
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 2
Q1. Prove, for a given graph G = (V, E), that the following are equivalent:
(i) G is a tree;
(ii) G contains no cycles and |E| = |V |
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 1
Q1. Suppose that there are n people and n jobs. Each job must be assigned to exactly
one person, and each person does only one job. T
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 3
Q1. Suppose that A Rmn is of full row rank (m n), c, d Zm and c < d. Show that
any extreme point to the polyhedron
S = cfw_x Rn | Ax
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
Semester I (2014/2015) MA4254 Discrete Optimization Tutorial 2
Solution or hint to selected questions
Q1. Recall that a tree is a connected acyclic graph.
(i) (ii): Theorem 1.2 in lecture notes.
(