Course Objectives

• Increase literacy in the field of linear programming.
• Begin to do research in linear programming.
• Enhance technical communication skills.

Texts and Supplemental Materials

R.W. Cottle, J.-S. Pang, and R.E. Stone, The Linear Complementarity Problem, Academic Press, 1992.

S.J. Wright, Primal-Dual Interior Point Methods, SIAM, 1996.

H.J. Greenberg, Advanced Analysis in Linear Programming, Chapter 3, preliminary version.

I.S. Duff, A.M. Erisman, and J.K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, 1986.

M.C. Ferris and O.L. Mangasarian, Linear Programming with MATLAB, preliminary version.

D.G. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, 1984 (2nd ed.).

S.G. Nash and A. Sofer, Linear and Nonlinear Programming, McGraw-Hill, 1996.

Assignments

Problem Sets: During the first half of the semester I will be assigning problem sets generally every week. This will taper off in the latter part of the semester in order to give you more time to focus on your projects. You are encouraged to discuss the homeworks with classmates; however, you should write up the solutions separately.

Presentation: Halfway through the semester, each student will give a 30 minute oral presentation on a topic to be agreed upon between the student and the instructor. The educational goal of this assignment is for you to delve deeper into a topic and extract sufficient understanding to communicate the subject to your peers in a relatively short amount of time. One approach you may want to take is to select a recent journal article and present it to the class. However, I am open to other ideas.

Project: Each student will do a term project. The educational goal of this project is for you to apply your now extensive mathematical knowledge to a nontrivial problem, and ultimately to move you toward the goal of doing independent research. Your project does not have to entail original research; however, it should involve reading several journal articles. There are four deliverables you must meet for your project:

1. Project proposal: You must turn in a written project proposal by the end of February. This proposal describes what you plan to do for your term project. It should also include an annotated bibliography of at least three journal articles.
2. Preliminary project report: About three weeks prior to the end of the semester, you must hand in a draft of your project report. This draft should be as complete as possible given the current state of your project. In particular, your draft report should contain an abstract, introduction, and background section which are essentially complete. It should also describe any results you have finished and indicate the state of the outstanding unresolved issues.
3. Oral presentation: During the last week (possibly two) of the semester, you will give a 30-minute presentation of your project.
4. Final report. You final project report will be due the last day of class. NO LATE REPORTS WILL BE ACCEPTED (unless prior arrangements are made).

Here are some sample project ideas:

• Find a nontrivial real-world problem and model and solve it as a linear program or linear complementarity problem.
• Do a literature search on recent advances on some aspect of linear programming (for example, modern pivot selection mechanisms), summarize the current state of the art and identify some outstanding research questions.
• Implement and test an algorithm from the literature agreed to by the instructor.

Grading

Important Dates

February 28       Project proposal due
March 20-25       Spring Break--no classes
March 27-April 5  Student presentations
April 17          Draft project report due
May 1-May 8       Project presentations
May 8             Project report due

Topical Outline

1. Background
   1. Review of LP
   2. Matrix analysis
   3. Convex polyhedra
   4. Separation theorems
   5. Theorems of the alternative
   6. Matrix factorizations
   7. Quadratic programming
   8. Newton's method
2. The linear complementarity problem
   1. Problem formulation
   2. Applications
   3. Existence and multiplicity of solutions
   4. Lemke's method
   5. Geometric interpretation of Lemke's method
   6. Generalizations
3. Interior point methods
   1. Primal-dual methods
   2. The central path
   3. Path-following methods.
   4. Potential reduction methods.
   5. Affine scaling methods.
   6. Mehrotra's predictor-corrector algorithm.
   7. Complexity theory
   8. Infeasible interior point methods.
   9. Semi-definite programming.
4. Sensitivity Analysis.
   1. Objective differentiation
   2. Rim response estimation
   3. Basic theory of compatibility
   4. Average prices
   5. Beyond the rim
5. Sparse Matrix Techniques
   1. Storage schemes
   2. Gaussian elimination for sparse matrices
   3. Pivotal strategies for sparse matrices