Assignment set 3 (final)
Deadline: December 5th
The following non-linear programming problems can be solved using various
software, for instance with MatLab (with optimization toolkit). The CD-ROM
with the book contains MPL (with non-linear solver Conopt), the non-linear
solver Lindo (http://www.lindo.com/) and an Exel solver.
You can also use the NEOS server to which you can send your optimization
problems for which a solution is then (usually) returned: http://wwwneos.mcs.anl.gov/neos/
FAQ on nonlinear optimization, containing other links to software:
http://www-unix.mcs.anl.gov/otc/Guide/faq/nonlinear-programming-faq.html
Free book about convex optimization:
Stephen Boyd & Lieven Vandenberghe: Convex Optimization
http://www.ee.ucla.edu/~vandenbe/cvxbook/
A free book containing all kinds of numerical algorithms, including some
optimization: William Press, Brian Flannery, Saul Teukolsky, William
Vetterling: Numerical recipes in C/Fortran.
http://www.nr.com/
From the following three problems two may be replaced by a (substantial)
optimization problem from your own field or interest.
1. Find the hexagon with the largest area such that any two vertices are
not more than one unit length apart. Compare your solution with an
equilateral hexagon.
2. Find the minimum of the following function:
f(x) = x12 + x22 + 2x32 + x42 - 5x1 - 5x2 – 21x3 + 7x4,
subject to the following constraints:
8 - x12 - x22 - x32 - x42 - x1 + x2 – x3 + x4  0,
10 - x12 - 2x22 - x32 - 2x42 + x1 + x4  0,
5 - 2x12 - x22 - x32 - 2x1 + x2 + x4  0.
3. A heavy ball rolls down a piecewise linear slope, starting at rest at
(0,1), via the points (xi,yi), I = 1, 2, 3, ending in point (1,0). Gravity is in
the negative y direction. Determine the position of the intermediate
points such that the total travel time of the ball is minimal.