ESE504 - Introduction to Optimization (Final Exam)
Fall Semester, 2015
M. Carchidi
Problem #1 (15 points) - Primal/Dual Problem
Suppose that the standard form linear programming problem,
min w = cT x (Objective Function)
s.t.
Ax = b (Constraints)
x0
(Sign

Introduction to Optimization
1
MIT and James Orlin 2003
What is Operations Research?
What is Management Science?
World War II : British military leaders asked scientists
and engineers to analyze several military problems
Deployment of radar
Management o

More Linear Programming Models
(modeling, modeling, modeling)
modified by MGS from the original slides by J. Orlin
Modeling is an art, not a science, and
only practice makes perfect.
So let us practice
1
Overview
Applications
personnel scheduling
radia

Animation of the Gauss-Jordan
Elimination Algorithm
Developed by James Orlin, MIT
1
Solving a System of Equations
x1
x2
x3
x4
1
2
-1
2
1
1
4
-1
2
1
-1
2
=
=
=
0
6
-3
To solve a system of equations, use Gauss-Jordan
elimination.
2
To solve the system of e

Chapter 4. The simplex algorithm
Putting
Linear Programs into standard form
Introduction to Simplex Algorithm
J. Orlin and MIT
1
Overview of Lecture
Getting
an LP into standard form
Getting an LP into canonical form
Optimality conditions
Improving a

BASIC FEASIBLE SOLUTIONS
CONTENTS
Polyhedral Sets and Polytopes
Basic Feasible Solutions and Polytopes
Reference: Chapters 2 and 3 in BJS book.
Hyperplanes
A hyperplane in En generalizes the notion of a straight line in
E2 and the notion of a plane in E

An Example of Two Phase Simplex Method
AdvOL @McMaster, http:/optlab.mcmaster.ca
February 2, 2009.
Consider the following LP problem.
max
s.t.
z = 2x1 + 3x2 + x3
x1 + x2 + x3
2x1 + x2 x3
x2 + x3
x1 , x 2 , x 3
40
10
10
0
It can be transformed into the

Simplex Method Continued
1
Next Lectures
Review
of the simplex algorithm.
Formalizing the approach
Degeneracy and Alternative Optimal Solutions
Is the simplex algorithm finite? (Answer, yes,
but only if we are careful)
2
LP Canonical Form =
LP Standar

Assignment Problems
Example: Machineco has four jobs to be completed.
Each machine must be assigned to complete one job.
The time required to setup each machine for completing
each job is shown in the table below. Machineco wants
to minimize the total set

University Of Pennsylvania
Department of Electrical and Systems Engineering
ESE504 Optimization Theory & Analysis (Course Outline)
Instructor: Dr. Michael A. Carchidi
-Textbook:
1.)
Introduction to Linear Optimization by Dimitris Bertsimas
and John N. Tsi

Introduction to Integer Programming
Integer
programming models
1
MIT and James Orlin 2003
A 2-Variable Integer program
maximize
3x + 4y
subject to
5x + 8y 24
x, y 0 and integer
What
is the optimal solution?
2
MIT and James Orlin 2003
The Feasible Region

Integer Programming 2
Branch
and Bound
Read Hillier, ch. 12
1
MIT and James Orlin 2003
Overview of Techniques for Solving
Integer Programs
Enumeration Techniques
Complete Enumeration
list all solutions and choose the best
Branch and Bound
Implicitly

A brief review of Linear Algebra and
Linear Programming Models
Developed by James Orlin
1
Review of Linear Algebra
Some
elementary facts about vectors and
matrices.
The Gauss-Jordan method for solving
systems of equations.
Bases and basic solutions and

Applications
A Financial
Model
an investment model over multiple time
periods
goal: optimize the total revenues at the end of
the time horizon
1
A Financial Problem
Sarah has $1.1 million to invest in five different
projects for her firm.
Goal: maximiz

ESE504 - Introduction to Optimization (Exam #2)
Fall Semester, 2016
M. Carchidi
Instructions:
1.) You must answer all of the following seven (7) problems given below. Note
that if you answer all 7 problems, 10 extra-credit points are possible.
2.) You mus

ESE504 - Introduction to Optimization (Exam #2)
Fall Semester, 2016
M. Carchidi
Instructions:
1.) You must answer all of the following seven (7) problems given below. Note
that if you answer all 7 problems, 10 extra-credit points are possible.
2.) You mus

ESE504 - Introduction to Optimization (Homework #5)
Fall Semester, 2016
M. Carchidi
Problem #1 (25 points) - An Antisymmetric Matrix
Consider the following LP problem
max z = bT x (Objective Function)
s.t.
Ax b (Constraints)
x0
(Sign Restrictions)
in whic

ESE504 - Introduction to Optimization (Homework #4)
Fall Semester, 2016
M. Carchidi
Problem #1 (15 points) - The Eciency/Ineciency of the Simplex
The simplex method is a typically ecient algorithm for solving linear programming problems. However, there is

ESE504 - Introduction to Optimization (Exam #2)
Fall Semester, 2016
M. Carchidi
Instructions:
1.) You must answer all of the following seven (7) problems given below. Note
that if you answer all 7 problems, 10 extra-credit points are possible.
2.) You mus

ESE504 - Introduction to Optimization (Homework #6)
Fall Semester, 2016
M. Carchidi
Problem #1 (15 points) - An Integer Programming Problem
Determine (without the use of a computer) the optimal solution to the
following MIP problem.
max
s.t.
z = 3x1 + x2

Linear Optimization Theory & Analysis
(ESE 504)
Michael A. Carchidi
October 16, 2016
Chapter 3 - The Geometry Behind Linear Programming
The following notes are based on the two textbooks entitled: Introduction to
Linear Optimization by Dimitris Bertsmias

Linear Optimization Theory & Analysis
(ESE 504)
Michael A. Carchidi
November 25, 2015
Chapter 5 - Duality Theory
The following notes are based on the two textbooks entitled: Introduction to
Linear Optimization by Dimitris Bertsmias and John N. Tsitsiklis

Linear Optimization Theory & Analysis
(ESE 504)
Michael A. Carchidi
November 3, 2015
Chapter 4 - The Simplex Method
The following notes are based on the two textbooks entitled: Introduction to
Linear Optimization by Dimitris Bertsmias and John N. Tsitsikl

Animation of the Gauss-Jordan
Elimination Algorithm
Developed by James Orlin, MIT
1
Solving a System of Equations
x1
x2
x3
x4
1
2
-1
2
1
1
4
-1
2
1
-1
2
=
=
=
0
6
-3
To solve a system of equations, use Gauss-Jordan
elimination.
2
To solve the system of e

Non Linear Programming 1
Nonlinear Programming (NLP)
Modeling Examples
1
MIT and James Orlin 2003
Linear Programming Model
Maximize c1 x1 c2 x2 . cn xn
subject to
a11x1 + a12 x 2 + . +a1n xn b1
a 21x1 + a 22 x 2 + . +a 2n xn b 2
M
M
M
am1 x1 + a m 2 x 2

Nonlinear Programming Theory,
Part 2 (revised)
1
MIT and James Orlin 2003
Difficulties of NLP Models
Linear Program:
Nonlinear
Programs:
2
MIT and James Orlin 2003
Graphical Analysis of Non-linear programs
in two dimensions: An example
Minimize
subject to

S O L V E S U M M A R Y
MODEL pro OBJECTIVE z
TYPE MIP DIRECTION MINIMIZE
SOLVER CPLEX FROM LINE 106
* SOLVER STATUS 1 Normal Completion
* MODEL STATUS 1 Optimal
* OBJECTIVE VALUE 21.2700
RESOURCE USAGE, LIMIT 4.087 1000.000
ITERATION COUNT, LIMIT

S O L V E S U M M A R Y
MODEL pro OBJECTIVE z
TYPE MIP DIRECTION MAXIMIZE
SOLVER CPLEX FROM LINE 106
* SOLVER STATUS 1 Normal Completion
* MODEL STATUS 1 Optimal
* OBJECTIVE VALUE 34.6900
RESOURCE USAGE, LIMIT 0.109 1000.000
ITERATION COUNT, LIMIT