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Discrete formulation

What are the next starts i should choose to reduce gemstones type difference, the discrete optimization point of view.

Let us consider the gemstones as a vector y = (y₁, y₂, y₃, y₄) in N⁴, where the components respectively respresent the number of margonite gemstones, stygian gemstones, torment gemstones, titan gemstones.

Then, we consider successful runs by the following functions going from N⁴ to N⁴:

f₁(x) = x + k(1, 2, 3, 4)

f₂(x) = x + k(4, 1, 2, 3)

f₃(x) = x + k(3, 4, 1, 2)

f₄(x) = x + k(2, 3, 4, 1)

Where k = 2 in Hard Mode (HM), and where the functions respectively represent start city, start veil, start gloom, start foundry.

Now, let g be a finite number n_i of composition of functions f_i, (i in {1,...,4}). We consider g to be solution if it is the minimal number of composition which gives the minimum possible difference of gemstones. This gives the following minimization problem:

g^* = argmin_g (max_{(i, j)} |g_i(y) - g_j(y)| + d||n||₁), where d is small.

Let us set h_(i,j)(y) = |g_i(y) - g_j(y)|. This is a polyhedral function with respect to n:

h_(i,j)(n, y) = |y_i - y_j + k<D_(i,j),n>|

where <., .> is the usual scalar product.

Continuous setting

Let us now consider z in R. The discrete problem is equivalent to this one:

min_z z + <n, 1>

under:

z >= y_i - y_j + k<D_(i,j),n>

z >= y_j - y_i + k<D_(j,i),n>

for all couple (i,j) in {(1, 2), (2, 3), (3, 4), (1, 3), (2, 4), (1, 4)}

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