|12/02/09|| Midterm#2 on Dec. 2nd
|11/20/09|| Assignment 10 is due on Dec 2nd .
|11/18/09|| Lecture 20 Notes
|11/13/09|| Homework Assignment 9 is due next friday (11/20) and late on 11/30, the monday after thanksgiving.
|11/06/09|| Homework Assignment 8 is due next friday (11/13) and late on the subsequent wednesday.
|11/4/09|| Lecture 16 Notes (handwritten notes).
|11/2/09|| Lecture 15 Notes.
|10/29/09|| Assignment 7 is due Nov 4th (wednesday)..
|10/26/09|| Lecture 14 Notes.
|10/22/09|| Assignment 6 is due on the 28th of october (wednesday).
|10/21/09|| Lecture 13 Notes.
|10/19/09|| Post-midterm review (grades given out), rehash of Lecture#12.
|10/14/09|| First midterm
|10/12/09|| Preparation for Midterm: come prepared with questions.
|10/07/09|| Lecture #12: Handwritten notes. Copies available from ECOT 624.
|10/05/09|| Assignment#5 is here (due 10/12, late 10/19).
|10/05/09|| Lecture #11 slides .
|09/29/09|| Lecture #10 slides , matlab codes: browse, download tarball
|09/28/09|| Lecture #9 slides , matlab codes: solveLeastSquares , expt1 , expt2 .
| 09/21/09|| Lecture #8 slides .
| 09/18/09|| Assignment #3 here is due on the 28th Sept. (one week late: 5th oct).
| 09/16/09 || Lecture #7 slides ,
Chapter 9 of Venderbei's book available online through Springer.
| 09/14/09 || Lecture #6 slides
| 09/09/09 || Lecture #5 slides and homework assignment #2 (due 09/16/2009, one week late: 09/25/2009).
| 09/09/09 || Lecture #5 Handout #3 .
| 09/02/2009 || Lecture #4 slides and homework assignment #1 (due 09/09/2009, one week late: 09/16/2009).
| 08/31/2009 || Posted lecture#3 slides
| 08/26/2009 || Posted lecture#2
slides (passwords given out in class).
| 08/24/2009 || Posted lecture#1
slides and Handout#1
(passwords given out in class).
- 6:45 p.m. @ ECCR 116.
Tuesdays/Fridays 2:00- 3:00 p.m. @ECOT 831, or by appointment.
This course presents the theory and applications of Linear
, arguably one of the most important optimization
problems in applied mathematics and engineering. The Simplex
algorithm to solve linear programs is widely regarded as one among the "top
of the 20th century. The theory of linear
programming is used in almost all fields of engineering including
operations research, machine learning, control system
design, scheduling, computer vision and so on. It forms the
numerous approaches to solving hard combinatorial optimization problems
through randomization and approximation.
The primary goals of CSCI 5654 will be
- Understand the basic theory behind LP, algorithms to solve LPs,
and basics of (mixed) integer programs.
- Understand important and emerging applications of LP to economic
problems (optimal resource allocation, scheduling
problems), machine learning (SVM), control
design (finite horizon optimal control, dynamic programming), formal
verification (ranking functions), and so on.
At the end of the course, the successful student will be able to cast
various problems that may arise in her research as optimization
problems, understand the cases where the optimization problem will be
linear, choose appropriate solution methods and interpret results
appropriately. This is generally a powerful tool in many research areas.
Topics that will be covered include:
- Linear Programs: Simplex algorithm and its
complementary slackness; Geometry of linear programs; Network flow
- Integer Linear Programs:
Branch & Bound, Gomory cuts, approximation/randomization through LP.
- Applications: Optimal
allocation of resources, control design, verification, regression,
relaxations of hard combinatorial problems to LP.
- Beyond LP: Convexity,
Quadratic Programming, Semi-Definite Programming.
The choice of topics may vary depending on available time
and student feedback.
The course will feature
weekly homework assignments, and two midterm quizes. The
final grade will be determined as follows:
|Class Participation (*)
(*) Class participation includes responding to in-class questions,
asking questions in class, making critical comments about the material
in class and volunteering
to lead a mini-presentation on a topic of interest (towards the end of
the course). It is expected that all the students will earn full
credits for class participation.
All exams and quizzes will be held in class.
The overall grade will be determined by adding up the homework
assignment and midterm scores in the ratio shown in the table
above. Your grade will be determined by the total score
(normalized to 100):
|Total Score range
|Expected Grade Range
| 75 - 100
| A- to A+
| 60 - 75
| B- to B+
| >= 45
| < 45
homework assignments are strictly non-collaborative. However, you are
permitted (and even encouraged) to discuss solution
the instructor or other people who are enrolled in the course. Any such
help or discussion must be
at the beginning of your assignment. Such
an acknowledgement will not incur any penalties.
The actual assignment itself must be written
entirely on your own
without outside assistance. Copying
directly or indirectly (consulting) from unauthorized
sources (solution manuals, on line web pages, discussion forum and so
on) is prohibited. If in doubt, ask the instructor. Looking at another
or solution is strictly prohibited.
Soliciting help on assignments from people or sources other than
the instructor and students registered for this course is considered
unauthorized assistance and hence a violation of the CU honor
code. Once again, if you have any doubts in this regard, ask the
Note: The CU Boulder honor code is available online HERE! You are
expected to be aware of and to follow the honor code.
by Vasek Chvatal, W.H. Freeman publisher, 1983.
Topics not covered in Chvatal will be covered by notes and handouts.
(todo for instructor: place this in reserve).
Linear Programming by R.J. Vanderbei (available online
through the CU library).
Combinatorial optimization: Algorithms & Complexity, by
Christos Papadimitrou and Ken Steglitz
Linear Programming: Theory
and Applications by G.B. Danzig and M.K. Thapa.
Integer and Combinatoral Optimization: Nemhauser & Wosley are also
good references (not recommended as texts).
Convex Optimization by Boyd
& Vandenbreghe is a good reference for the more general area of
Announcements and Handouts
| Lecture || Topics Covered || Slides + Handouts |
| || ||
|20 || Cutting Plane Methods: (Chvatal-) Gomory cuts. || Lecture Notes
|19 || Integer Linear Programming: NP completeness, examples, branch-and-bound method. || Vanderbei's book chapter 23.
|17 & 18|| Network Simplex: Transshipment problem. Initialization Phase. Analysis of Network Simplex Problems.|| Chvatal Chapter 18, and Vanderbei's book Chapter 14.
|16|| Polar cones. Network Simplex: Transshipment problem and Feasible Spanning Tree Solutions. || Handwritten notes .
|15|| Double Description of Polyhedra, Geometric view of degeneracy || Lecture Notes .
|14|| Farkas' Lemma (theorems of the alternative) and Minkowski-Weyl Theorem || Lecture Notes.
|13|| Convex polytopes, polyhedra, vertices, basic feasible solutions and correspondence to Simplex || Lecture Notes.
|12|| Geometry of Simplex-1: basic properties of linear spaces, affine spaces and cones (rehash
of lecture 12, because of midterm interruption.)||
|12|| Geometry of Simplex-1: basic properties of linear spaces, affine spaces and cones.|| handwritten note (to be typeset soon).
|11|| Classification problems: classification as LP, dual form of linear-separable classification problem, support vectors, pitfalls using a linear objective function, margins. Also discussed optimal margin classifier using SVMs (not in syllabus).|| slides.
|10 || Application of norm(penalty) minimization problems: regularization, penalty function approximation, signal smoothing (denoising), linear regression || slides , matlab codes: browse, download tarball
| 9 || Application module #1: Norm minimization problems (L1,L2, Linfty), unconstrained least squares, solving linear systems with noise || slides , matlab codes: solveLeastSquares , expt1 , expt2 .
| 8 || Duality for general form Simplex. Revision of basic concepts.|| slides
| 7 || Dual simplex: connections between primal/dual dictionaries, general form simplex|| slides
| 6 || Duality: revisited strong duality proof, complementary slackness, dual variables as shadow (margin) costs. || slides
| 5 || Dual Problem: Introduction, Dual Variables, Relations between dual and primal formulations. || slides and handout
| 4 || Pitfalls: Degeneracy, Cycling, Bland's Rule (w/o proof), Klee-Minty Cubes and Complexity. || Slides
| 3 || Pitfalls: Unbounded instances, Initialization Phase Simplex and properties ||
| 2 || Basic Simplex: Dictionaries, Basic Variables, Basic Feasible Solution, Pivoting. ||
| 1 || Overview: Linear Programming, Topics, Example Applications ||
Slides and Handout #1 .
Assignments will be posted here. You are allowed a grace period of upto
one week for 5 assignments of your choice.
- Homework Assignment 10 is due on Dec 2nd
- Homework Assignment 9 is due 11/20 and late on 11/30.
- Homework Assignment 8 is due next friday (11/13) and late on the subsequent wednesday.
- Assignment 7 is due Nov 4th (wednesday)..
- Assignment 6 is due on the 28th of october.
- Homework Assignment #5 here is due on monday 12th October (one week late: 19th Oct).
- Homework Assignment #4 here is due on monday 5th October (one week late: 12th Oct).
- Homework Assignment #3 here is due on monday 28th Sept. (one week late: 5th oct).
- Homework Assignment #2 ( here ) will be due on 09/16/09.
- Homework Assignment #1 ( here ) will be due on 09/09/09.
Will post interesting links and papers here.