LAGRANGIAN RELAXATION AND PEGGING TEST FOR LINEAR ORDERING PROBLEMS( SCOPE (Seminar on Computation and OPtimization for new Extensions))

We develop an algorithm for the linear ordering problem, which has a large number of ap- plications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the La- grangian relaxation of a binary integer linear programming formulation of the problem. Since the number of the constraints is proportional to the third power of the number of items and grows rapidly, we propose a modied subgradient method that temporarily ignores a large part of the constraints and gradually adds constraints whose Lagrangian multipliers are likely to be positive at an optimal multiplier vector. We also propose an improvement on the ordinary pegging test by using the problem structure.

[1]  Arthur M. Geoffrion,et al.  Lagrangian Relaxation for Integer Programming , 2010, 50 Years of Integer Programming.

[2]  Robert Sedgewick,et al.  Algorithms in C , 1990 .

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

[4]  Santosh S. Vempala,et al.  Fences Are Futile: On Relaxations for the Linear Ordering Problem , 2001, IPCO.

[5]  John G. Kemeny,et al.  Mathematical models in the social sciences , 1964 .

[6]  Kathryn E. Pedings,et al.  A minimum violations ranking method , 2012 .

[7]  Samuel Fiorini,et al.  Facets of the linear ordering polytope: A unification for the fence family through weighted graphs , 2006 .

[8]  John G. Kemeny,et al.  Mathematical models in the social sciences , 1964 .

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

[10]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[11]  Gerhard Reinelt,et al.  The Linear Ordering Problem , 2011 .

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

[13]  Fred W. Glover,et al.  An Experimental Evaluation of a Scatter Search for the Linear Ordering Problem , 2001, J. Glob. Optim..

[14]  J. Mitchell,et al.  Solving Linear Ordering Problems with a Combined Interior Pointtsimplex Cutting Plane Algorithm * , 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]  Stefan Chanas,et al.  A new heuristic algorithm solving the linear ordering problem , 1996, Comput. Optim. Appl..

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

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

[20]  Rafael Martí,et al.  Variable neighborhood search for the linear ordering problem , 2006, Comput. Oper. Res..

[21]  W. Cook,et al.  On the Minimum Violations Ranking of a Tournament , 1986 .

[22]  Eranda Çela,et al.  The quadratic assignment problem : theory and algorithms , 1999 .

[23]  M. Patriksson,et al.  Conditional subgradient optimization -- Theory and applications , 1996 .

[24]  John E. Mitchell,et al.  Solving real-world linear ordering problems using a primal-dual interior point cutting plane method , 1996, Ann. Oper. Res..

[25]  Abilio Lucena,et al.  Lagrangian heuristics for the linear ordering problem , 2004 .

[26]  V. Deineko,et al.  The Quadratic Assignment Problem: Theory and Algorithms , 1998 .

[27]  Irène Charon,et al.  A branch-and-bound algorithm to solve the linear ordering problem for weighted tournaments , 2006, Discret. Appl. Math..

[28]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[29]  Irène Charon,et al.  A survey on the linear ordering problem for weighted or unweighted tournaments , 2007, 4OR.