Math. 655 Course Outline

As of August 14, this page is still under development.  Further details will be added soon.  (I promise.)  RBK
This outline is a tentative guide, and will be updated as the course progresses.  Time will be allocated as appropriate for student presentations and discussion.

Note: Some of the material referenced below may be copyrighted, and hence is not generally available by clicking on it. Please email me,, and I will email you either a Postscript or a PDF copy of the appropriate excerpts, according to the ``fair use" provision of the copyright law.

Home page for the course
Bibliography for the course in PDF format
Outline for the somewhat related course from Fall, 2003
Bibliography for the 2003 course in PDF format
Recommended references for the first part of the course:

  1. The portions of our 2005-2006 Math. 555-556 manuscript dealing with interval arithmetic and on global optimization.  This includes §1.3, §2.5, §3.4.5, §4.3.9, §8.4, and §9.3, and §9.5.3.
  2. The book Rigorous Global Search: Continuous Problems (R. B. Kearfott, Kluwer Academic Publishers, 1996).  I have a couple of spare copies I can lend during the course.
Description Explanation / References / Projects
Overview of the course and of the inventory of tools 
  1. A review of interval arithmetic and a discussion of its practicalities. 
  2. Review of Matlab and INTLAB
  3. Toying with the NEOS server
  • The writeup "A critical assessment of interval arithmetic practicalities."
  • The computing equipment and software page for this course
  • §9.5.3 of the 2005-2006 Math 555-556 text.
  • The NEOS server page
  • Discussion of Extended interval arithmetic The class notes An Example to Motivate Issues in Extended Interval Arithmetic
    The containment set (cset) proposal for extended interval arithmetic
    The C++ interval arithmetic standardization effort
    The ISL project: participation possibilities
  • The ISL web site:
  • The "mission" paper (available from the moodle course)
  • The "extended interval arithmetic inventory" (an Excel spreadsheet available from the moodle course)
  • Languages for modeling
  • GAMS
  • AMPL
  • MPS

  • Problems of interest are commonly expressed in these languages.
    Review of branch and bound processes for global optimization Section 9.5.3 of our 2005-2006 Math. 555-556 manuscript, available on the Moodle course for this course.
    A review of interval Newton methods for existence and uniqueness Section 8.4 of our 2005-2006 Math. 555-556 manuscript, available on the Moodle course for this course.
    epsilon-inflation Section 4.2 of Rigorous Global Search: Continuous Problems, R. B. Kearfott, Kluwer Academic Publishers, 1996.
    A review of necessary and sufficient conditions for a point to be a local optimizer
    • Statement of the general Kuhn-Tucker and Fritz John conditions
    • The unconstrained case
    • The linear equality constrained case
    • The linear inequality constrained case
    • The nonlinear case
    We will be using primarily excerpts from Chapter 3 of Gill, Murray, and Wright, Practical Optimization, Academic Press, 1981.  (Scans of a few selected pages are available on the Moodle course for this course).
    A possible research project in branch and bound methods Relationship of Necessary and Sufficient Conditions to Branch and Bound Algorithms, available on the Moodle course for this course.