413047-Underground-Commercial-Sex-Economy

1.4.7. Techniques for Solving the Linear Program
We use the non-commercial (LGPL2 license) linear programming solver lp_solve, written in ANSI C by
Michel Berkelaar, which is known to efficiently solve problems with up to 30,000 variables and 50,000
constraints.
1.5. Geometric Intuition
The picture that follows (Figure 1 A Geometric Representation of the Optimization Problem) attempts to
illustrate the problem at an intuitive level. Although the figure is just 2-dimensional (the axes shown in
black), in actuality, our problem is 56-dimensional (7 cities times 2 times 4 economic unknowns, namely
sex, drugs, guns, and other). One might imagine then that the two black axes shown in the figure are just 2
of the 56, say the UCSE in Dallas ( ) in 2003 (on the x-axis), and the UCSE in Miami ( ) in 2003 (yaxis).
We know from Sex Proxy Ratios analysis (see section 1.4.1. Sex Proxy Ratio Constraints) that the ratio of
UCSE in Dallas and the UCSE in Miami in 2003 is
and accounting for 20 percent slack in the estimation of this ratio via , we can assume that the ratio
lines in the range [1.68, 2.52]. Thus, the ratio constraint defines a feasible “wedge” (shown in green), the
region between
Given that we have 448 ratio constraints (each between two variables), in actuality there are 224 such
green wedges, all emanating from the origin in 56-dimensional space.
The semantics of our variables necessitates that only solutions in the all-positive quadrant be considered.
We depict this by showing the positive quadrant in orange. In 56 dimensions, the all-positive quadrant
comprises a tiny fraction of the entire space. Formally, this is done with the positivity constraints
in section 0.
Finally, we have the equations that arise from the Law of Cash Conservation. These are non-homogeneous
(since they each contain a non-zero constant term Z) and so they do not pass through the origin. There are
14 such non-homogeneous equations, each relating 4 variables; the figure shows just one of the 14, in
blue.
In principle, it is unlikely that the 14 LoCC equations will have a simultaneous solution, so the linear
program seeks to find the unique point that is both “feasible” (i.e., in the intersection of all the green
wedges) but also minimizes its divergence from the LoCC equations (i.e., minimizes its total distance to
the blue constraints). The quality of the solution is evaluated via E, the average distance from the solution
point to the 14 LoCC equations.
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