IEE 598 - 9 (II.3.6) Lagrangian Relaxation
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Mar. 04, 2011
Lagrangian Relaxation
Recall that for LP relaxation, we dropped the integrality constraint. For an optimization problem zIP = maxcfw_cx : Ax b, Dx d

IEE 598 - 9 (II.3.6) Lagrangian Relaxation
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Mar. 04, 2011
Lagrangian Relaxation
Recall that for LP relaxation, we dropped the integrality constraint. For an optimization problem zIP = maxcfw_cx : Ax b, Dx d

IEE 598 - 8 (III.1) Integral Polyhedra
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 25, 2010
Integral Polyhedron Denition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point.
Zhang
IEE 376 Int

IEE 598 - 8 (III.1) Integral Polyhedra
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 25, 2010
Integral Polyhedron Denition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point. Proposition: A no

IEE 598 - 8 (III.1) Integral Polyhedra
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 24, 2011
Integral Polyhedron Denition: A nonempty polyhedron P is said to be integral if each of its nonempty faces contains an integral point. Proposition: A no

IEE 598 - Lecture 11 (I.5) Complexity Theory
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 24, 2009
Complexity Theory
How to measure an efcient algorithm? What kind of problems are hard and what are easy?
Zhang
IEE 376 Introduction to OR
2/1
Deci

IEE 598 - 6 (I.4) Review: Polyhedron Theory
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 11, 2010
Review: Polyhedral Theory
Denition: H = cfw_x Rn : ax = b is called a hyperplane. Denition: S = cfw_x Rn : ax b is called a (closed) half space. De

IEE 598 - 6 (I.4) Review: Polyhedron Theory
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 11, 2010
Review: Polyhedral Theory
Denition: H = cfw_x Rn : ax = b is called a hyperplane. Denition: S = cfw_x Rn : ax b is called a (closed) half space. De

IEE 598 - 5 (I.1) Review Linear Algebra
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 09, 2010
Review: Linear Algebra
Denition: is said to be a linear combination of x1 , x2 , ., xk Rn if = x x i R, i = 1, ., k. Denition: is said to be an afne co

IEE 598 - 5 (I.1) Review Linear Algebra
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 09, 2010
Review: Linear Algebra
Denition: is said to be a linear combination of x1 , x2 , ., xk Rn if = x x i R, i = 1, ., k. Denition: is said to be an afne co

IEE 598 - 4. (I.1) Alternative Formulations
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 04, 2010
Alternative Formulations
Example: (Uncapacitated) facility location problem There are a set N = cfw_1, ., n of potential facility locations and a s

IEE 598 - 4. (I.1) Alternative Formulations
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 04, 2010
Alternative Formulations
Example: (Uncapacitated) facility location problem There are a set N = cfw_1, ., n of potential facility locations and a s

IEE 598 - 3. (I.1, II.4.2) IP and LP
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 02, 2010
Relaxation Denition: (R) : z = maxcfw_c(x) : x T is called a relaxation of R (P) : z = maxcfw_f (x) : x S if P
1 2
S T, c(x) f (x), x S.
Proposition: z z

IEE 598 - 3. (I.1, II.4.2) IP and LP
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Feb. 02, 2010
Relaxation Denition: (R) : z = maxcfw_c(x) : x T is called a relaxation of R (P) : z = maxcfw_f (x) : x S if P
1 2
S T, c(x) f (x), x S.
Proposition: z z

IEE 598 - Lecture 2 (I.1) Modelling with Integer Variables
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Jan. 21, 2010
Why do we need integer variables?
Zhang
IEE 376 Introduction to OR
2 / 17
Why do we need integer variables?
When the variables are a

IEE 598 - Lecture 2 (I.1) Modelling with Integer Variables
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Jan. 21, 2010
Why do we need integer variables?
Zhang
IEE 376 Introduction to OR
2 / 12
Why do we need integer variables?
When the variables are a

IEE 598 - Lecture 1. Introduction (I.1)
Muhong Zhang
DEPARTMENT OF INDUSTRIAL ENGINEERING
Jan. 19, 2010
What is a mathematical programming?
Mathematical Programming is the use of mathematical models, particularly optimizing models, to assist in taking dec