Large Neighborhood Search

Abstract

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Geometric and LP-based heuristics for angular travelling salesman problems in the plane

“…In the objective function (20) costs c uvt are taken into account for the pair of consecutive edges {u, v}, {v, t} ∈ E, u = v = t, if both edges {u, v} and {v, t} are contained in the tour. Equations (21) are the degree constraints ensuring that each vertex is visited exactly once, (22) are the well-known subtour elimination constraints and, finally, (23) are the integrality constraints on the edge variables. In comparison to the standard model for the TSP by Dantzig et al [7] only the objective function is changed.…”

“…Our approach works as follows. Based on this ILP, we relax the integrality constraints (23) and (26) and the subtour elimination constraints (22) and solve the obtained LP (obviously having polynomial size) to optimality. Let us denote by x ′ e the obtained solution value of the x-variable for each edge e ∈ E. To construct a partial solution we first set x e = 0 for all e ∈ E and choose a rounding parameter ρ.…”

“…The magnifying glass heuristic can be understood as a large neighborhood search, where a (starting) solution is partially destroyed and repaired by corresponding operators; a process, which is iteratively repeated until a local optimum is reached (see [22]). The underlying idea of our approach works as follows: We consider a subset S ⊂ V of vertices and remove all edges adjacent to S from the tour.…”

“…These constraints ensure that the edges {a i , b i }, 1 ≤ i ≤ τ , representing the paths P i are contained in the tour. To solve this ILP to optimality, we adopt the integral approach proposed by Aichholzer et al in [2], i. e. we relax the subtour elimination constraints (22), solve the remaining ILP to integrality, add one subtour elimination constraint for every generated subtour and repeat the whole process until the solution consists of just one tour T ′ . This cycle can then be easily transformed into a new tour T ′ in the original graph with z(T ′ ) ≤ z(T ).…”

Ejection chain moves for automatic neighborhood synthesis in constrained cardinality‐minimization problems

“…Finally, such links are used to identify a set of entities (and, in turn, the corresponding variables). The model-based local search moves are then obtained by employing the well-known destroy-and-repair paradigm (Pisinger and Ropke, 2010), where a "destroy action" consists in making free the variables identified according to the above relationships. On the other hand, the "repair action" is performed by using an off-the-shelf black-box MIP (or CP) solver to obtain a new feasible solution.…”