By Ascheuer N., Junger M., Reinelt G.

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**Additional info for A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints**

**Sample text**

Basic SOFM training goes through the following steps and a variety of variants of SOFM can be found in [49]. Computational Intelligence in Clustering Algorithms 35 1. Define the topology of the SOFM; Initialize the prototype vectors mi (0), i = 1, . . , K randomly; 2. e. J = arg minj { x − mj }; 3. Update prototype vectors mi (t + 1) = mi (t) + hci (t)[x − mi (t)], where hci (t) is the neighborhood function that is often defined as hci (t) = rc −ri 2 ), where α(t) is the monotonically decreasing learning α(t) exp( − 2σ 2 (t) rate, r represents the position of corresponding neuron, and σ(t) is the monotonically decreasing kernel width function, or hci (t) = α(t) if node c belongs to neighborhood of winning node J 0 otherwise 4.

Activate layer F2 by choosing node J with the winner-takes-all rule TJ = maxj {Tj }; 4. Compare the expectation from layer F2 with the input pattern. If ρ ≤ |x ∩ WJ21 |/|x|, then go to step 5a, otherwise go to step 5b. 5. a Update the corresponding weights for the active node as 21 x∩WJ (old) and WJ21 (new) = x ∩ WJ21 (old); WJ12 (new) = β+|x∩W 21 J (old)| b Send a reset signal to disable the current active node by the orienting subsystem and return to step 3; 6. Present another input pattern, return to step 2 until all patterns are processed.

In [5], it is employed to determine a finite sequence of source points” {Qi } starting from PC , and a corresponding partition of Ω, {Ωi }. Each source point is the farthest visible point on ∂Ω from its predecessor in the sequence. The sequence of source points determines a partition of Ω into subdomains, such that each subdomain Ωi is the maximal region in Ω \ ∪i−1 j=1 Ωj which is visible from Qi . Then for P ∈ Ωi we have i−1 h(P ) = |P − Pi | + j=1 |Pj+1 − Pj |. For a domain with multiple singularity components, we employ N additional dimensions to discard the N singularity components {Ci , i = 1, .

### A Branch & Cut Algorithm for the Asymmetric Traveling Salesman Problem with Precedence Constraints by Ascheuer N., Junger M., Reinelt G.

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