The inequalities for all small rhombi are satisfied, for example 6+4 ≥ 10+0 and 10+4 ≥ 6+7 and 10+6 ≥ 12+4.
A weighting w ∈ R
36 is a vector of real numbers. For a given weighting w, the w-weight weightw(h) of a
mountain h is defined as
weightw(h) = X
w(v) · h(v).
For a given weighting w, a mountain h is called w-optimal if there is no other mountain with a higher w-weight.
The set of mountains whose heights at the boundary nodes are as illustrated in Figure 1 forms a polytope
15. Your taks is to experimentally determine the number of the polytope’s vertices. This shall be done by
choosing random linear objective functions and maximising them within the polytope.
1. Write a function (in any programming language) that picks a random weighting w and writes to the disk
an .lp file whose optimal solution is a w-optimal mountain.
2. Write a program that calls the above function 1000 times to generate 1000 .lp files, calls glpsol on those
files and analyses the output. After this analysis, the program should output the set of distinct mountains
that were found.
3. For each mountain that was found, draw a picture analogous to Figure 2. Put them in a single .pdf file.
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