Lagrangian heuristics for the linear ordering problem

Two heuristics for the Linear Ordering Problem are investigated in this paper. These heuristics are embedded within a Lagrangian Relaxation framework and are started with a construction phase. In this process, some Lagrangian (dual) information is used as an input to guide the construction of initial Linear Orderings. Solutions thus obtained are then submitted to local improvement in an overall procedure that is repeated throughout Subgradient Optimization. Since a very large number of inequalities must be dualized in this application, Relax and Cut is used instead of a straightforward implementation of the Subgradient Method. Heuristics are tested for instances from the literature and also for some new hard to solve exactly ones. From the results obtained, one of the proposed heuristics has shown to be competitive with the best in the literature. In particular, it generates optimal solutions for all 79 instances taken from the literature. As a by-product, it also proves optimality for 72 of them.

[1]  J. Mitchell,et al.  Solving Linear Ordering Problems with a Combined Interior Point/Simplex Cutting Plane Algorithm , 2000 .

[2]  Gerhard Reinelt,et al.  A note on small linear-ordering polytopes , 1993, Discret. Comput. Geom..

[3]  Cid C. de Souza,et al.  Optimal rectangular partitions , 2003, Networks.

[4]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[5]  Gerhard Reinelt,et al.  A Cutting Plane Algorithm for the Linear Ordering Problem , 1984, Oper. Res..

[6]  Eberhard Girlich,et al.  New Facets of the Linear Ordering Polytope , 1999, SIAM J. Discret. Math..

[7]  Carlos Garc,et al.  A Variable Neighborhood Search for Solving the Linear Ordering Problem , 2001 .

[8]  J. F. Bonnans.  Optimisation numérique : aspects théoriques et pratiques , 1997 .

[9]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[10]  Ulrich Faigle,et al.  A Lagrangian relaxation approach to the edge-weighted clique problem , 2001, Eur. J. Oper. Res..

[11]  Kavindra Malik,et al.  A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships , 1994, Ann. Oper. Res..

[12]  Rafael Martí,et al.  Intensification and diversification with elite tabu search solutions for the linear ordering problem , 1999, Comput. Oper. Res..

[13]  Pierre Hansen,et al.  Variable Neighborhood Search , 2018, Handbook of Heuristics.

[14]  M. Laguna,et al.  Scatter search for the linear ordering problem , 1999 .

[15]  Gerhard Reinelt,et al.  The linear ordering problem: algorithms and applications , 1985 .

[16]  Martin Grötschel,et al.  Facets of the linear ordering polytope , 1985, Math. Program..

[17]  Jon Lee,et al.  More facets from fences for linear ordering and acyclic subgraph polytopes , 1994, Discret. Appl. Math..

[18]  Stefan Chanas,et al.  A new heuristic algorithm solving the linear ordering problem , 1996, Comput. Optim. Appl..

[19]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[20]  Marshall L. Fisher,et al.  The Lagrangian Relaxation Method for Solving Integer Programming Problems , 2004, Manag. Sci..

[21]  Abilio Lucena,et al.  Stronger K-tree relaxations for the vehicle routing problem , 2004, Eur. J. Oper. Res..

[22]  Fred W. Glover,et al.  A Template for Scatter Search and Path Relinking , 1997, Artificial Evolution.