Principles of Mathematics in Operations Research

Web material

Remark 5.4.6 Xj = minxSRn R(x) \j = maxl6r R(x)

s.t. s.t.

xTv\ = 0 xTvj+i = 0

xTVj-i = 0 xTvn = 0

Problems

5.1. Prove the following theorem.

Theorem 5.4.7 (Rayleigh-Ritz) Let A be symmetric, Ai < A2 < • • • < An.

Ai = min x^1 Ax, An = max x^1 Ax.

IMI=i llx||=i

5.2. Use

to show Theorem 5.3.1.

5.3. Let

A =

1

" 100

2 10

1 2 1

0 1 1

f(x 1 ,x 2 ) = -x\ + -x\ + 2xxx 2 + -x\ -x 2 + 19.

Find the stationary and boundary points, then find the minimizer and the
maximizer over — 4 < x 2 < 0 < x\ < 3.

Web material

http://bmbiris.bmb.uga.edu/wampler/8200/using-ff/sld027.htm

http://delta.cs.cinvestav.mx/~mcintosh/comun/contours/node8.html

http://delta.cs.cinvestav.mx/-mcintosh/oldweb/lcau/node98.html

http://dft.rutgers.edu/~etsiper/rrosc.html

http://econ.lse.ac.uk/courses/ec319/M/lecture5.pdf

http://employees.oneonta.edu/GoutziCJ/fall_2003/math276/maple/

Lesson_141.html

http://en.wikipedia.org/wiki/Cholesky_decomposition

http://en.wikipedia.org/wiki/Maxima_and_minima

http://en.wikipedia.org/wiki/Positive-semidefinite_matrix

http://en.wikipedia.org/wiki/Quadratic_form

http://eom.springer.de/b/b016370.htm

http://eom.springer.de/C/cl20160.htm