413047-Underground-Commercial-Sex-Economy

We implement these 56 lower-bound linear inequalities using
and
. This expresses the lower bound and positivity requirement at a scale
that is appropriate to the problem, which is necessary to avoid numerical instabilities in the lp_solve
linear program optimizer. In other words, we assumed that at least 0.1 percent of all currency in
circulation is spent on each category, for each city and year.
1.4.6. An Objective Function Based on the Law of Cash Conservation
The previous sections (1.4.1. Sex Proxy Ratio Constraints to 1.4.5. Positivity Constraints) described the set
of
linear inequalities that must be simultaneously satisfied. Assuming the feasible set
is non-empty (i.e., there is at least one assignment of values to the 56 unknowns which satisfies these 448
inequalities), then there are likely to be infinitely many solutions. This is simply because the nonempty
intersection of these 448 halfspaces will, in general, be a convex subspace of . To specify a unique
solution from within the (infinite) feasible set, we must further specify an objective function that we
wish to minimize over the feasible set. In this setting, we take to be the average relative divergence from
the Conservation Law:
with
Note that:
1. E is non-negative, since it is the sum of 14 non-negative quantities.
2. if and only if the following 14 LoCC-based equalities hold precisely:
3. Each LoCC-based equality that fails to hold precisely contributes to the absolute value of the
relative discrepancy between its left and right hand sides (over 14).
By minimizing then, we effectively seek to find the unique point within the feasible set (i.e. a point
which satisfies the 448 linear inequalities) that also minimizes the absolute normalized discrepancy
between the left and right hand sides of the 14 equalities implied by the Law of Conservation of Cash.
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