Topic no. 
Description 
Explanation / References / Projects 
1. 
Overview of the course and a review of interval arithmetic 
Kreinovich's web
page,
Kearfott's Euromath
Bulletin article,
Sun's interval compiler
introduction ,
Kearfott's preprint
page and
Sun's
Interval Arithmetic Programming Reference

2. 
A review of interval Newton methods 
preprint of Kearfott's Encyclopedia
of Optimization article and
§1.5 of Rigorous Global Search: Continuous Problems (email
rbk@louisiana.edu
for a copy.)

3. 
A review of necessary and sufficient conditions for global optimization 
§3.3 and §3.4 of Gill / Murray / Wright,
§5.2.5 of Rigorous Global Search: Continuous Problems (email
rbk@louisiana.edu
for a copy.)

4. 
Hansen's technique for substituting points for some intervals in the
derivative matrices 
§1.3 of Rigorous Global Search: Continuous Problems (email
rbk@louisiana.edu
for a copy.), also
slopebased existence and uniqueness in §1.5 of Rigorous Global
Search: Continuous Problems (email rbk@louisiana.edu
for a copy.)

5. 
Interval Newton methods applied to the Fritz / John equations 
Examples of interval Newton methods with equality, inequality, and bound
constraints; discussion of the Lagrange multipliers;
E. Hansen and W. Walster, "Bounds for Lagrange Multipliers and Optimal
Points," Comput. Math. Appl.25 (10), pp. 59 ff
Project: Investigate practical behavior of estimates
for Lagrange multipliers in local computational existence / uniqueness.

6 
Interval Newton Methods and nonSmooth problems 
Chapter 6 of Rigorous Global Search: Continuous Problems, and an
excerpt from
I. Zang, "A SmoothingOut Technique for MinMax Optimization, Mathematical
Programming 19 (1), pp. 6167 (1980) (email rbk@louisiana.edu
for a copy.)
Project: Compare the method of representing breakpoints
in Zang to that in "Rigorous Global Search." Which method
gives smaller intervals? How would we compute interval slopes for the Zang
representation?
Project: Compare verification processes obtained
from adding constraints to verification processes associated with handling
the nonsmooth problems directly.

7. 
Generalizations of interval Newton Methods 
Project: Investigate the "higher order" interval
Newton method described in M. Berz and J. Hoefkens, "Verified HighOrder
Inversion of Functional Dependencies and
Interval Newton Methods, Reliable Computing
7 (5), pp. 379398, 2001. (email rbk@louisiana.edu
for a copy.) 
8. 
Extended interval arithmetic and optimization 
G. W. Walster, J. D. Pryce, and E. R. Hansen," ExceptionFree Arithmetic
on the Extended Reals," seminar notes from the Royal Military College of
Science, Cranfield, U.K. (email rbk@louisiana.edu
for a copy.)
We will be doing some experiments with interval Newton methods and extended
interval arithmetic, in particular, looking at the relationship to
necessary conditions for optima.
Project: In Sun's extended interval arithmetic, does
0 need to be an element of grad(f)(x)
for x to contain a critical point? (There are issues of domain
of definition and of smoothness.)

9. 
Singular systems and the topological degree 
§1.6 of Rigorous Global Search: Continuous Problems (email
rbk@louisiana.edu
for a copy.),
"Existence
Verification for Singular Zeros of Complex Nonlinear Systems,” SIAM
J. Numer. Anal. 38 (2), pp. 360379 (2000). (Copyright
SIAM.)
"Existence
Verification for Higher Degree Singular Zeros of Complex Nonlinear Systems,"
(joint with J. Dian), submitted to the SIAM
J. Numer. Anal.
"Verifying
Topological Indices for HigherOrder Rank Deficiencies," submitted
for the proceedings of the 2000 AMSIMSSIAM summer research conference
on Algorithms and their Complexity for Nonlinear Problems, July
16 to July 20, 2000, Mt. Holyoke College, edited by Christopher Sikorski.
Projects:
Implement computation of the topological index for
higherorder rank deficiencies (worth at least a publication)
Investigate degree computation in the nonsmooth
case.

10. 
Interval Newton methods and least squares: the augmented system 
M. Arioli, I. S. Duff, and P. P. M. de Rijk, "On the Augmented System Approach
to Sparse LeastSquares Problems," Numer. Math. 55 (1989),
pp. 667684.
Notes on implementing the augmented system (email rbk@louisiana.edu
for a copy.)

11. 
Least squares, a possible new paradigm 
Uncertainties in the data can be explicitly taken into account with interval
techniques
References:
A. Neumaier, "Solving Linear Least Squares Problems by Interval Arithmetic,"
Freburger IntervallBerichte 87, 6, pp. 3742.
A. Neumaier, "Linear Interval Equations," in Interval Mathematics 1985
(Lecture Notes in Computer Science no. 212), ed. K. Nickel, Springer
Verlag, 1985
Project: Implement the idea in the Neumaier reference
and see how it does with some realistic data sets.

12. 
Workshop on good programming style 
W.
S. Brainerd, C. H. Goldberg and J. C. Adams, Programmer's Guide to Fortran
90, third edition, Springer Verlag, New York, 1995.
We will cover the following:

Global versus local data (Proper use of modules, locality of reference,
ease and readability of subroutine calls; various examples will be given.)

Indentation and other readability considerations (loop indentation, ifthenelse
and case, choosing variable and label names, avoidance of "jumps")

Modularity (making subroutines into logical units, internal versus external
subroutines)

Machine efficiency versus readability and maintainability (human efficiency)

A discussion of userdefined data types

13. 
Workshop on mathematical writing 
"Plain Writing ... What's Needed and Why," Address given by I. E. Block,
SIAM Managing Editor, July 18, 1989 (Copies will be handed out in class.)
Also, Handbook
of Writing for the Mathematical Sciences by Nicholas J. Higham,
SIAM, Philadelphia, 1993 (in Dupré Library).

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