K. Hoffman and M. Padberg. LP-based combinatorial problem solving. Annals of Operations Research, 4:145--194, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one. 2. 3 Branch, Constrain, and Price Branch, constrain, and price is an implementation of branch and bound used most typically for solving problems that can be formulated as integer programs (in this context, the method is commonly referred to as branch, cut, and price) Early works such as [19, 20, 27] laid out the basic framework of BCP and since then, many implementations (including SYMPHONY and COIN BCP) have built on these preliminary ideas. In a BCP algorithm for solving an integer program, the bounding operation is performed using tools from linear programming and polyhedral theory. Since ....

K. Hoffman and M. Padberg, LP-Based Combinatorial Problem Solving, Annals of Operations Research 4 (1985/86), 145.

....to model as integer programs, have long remained challenging to solve in practice. The last two decades have seen tremendous progress in our ability to solve large scale discrete optimization problems. These advances have culminated in the approach that we now call branch and cut, a technique (see [45, 72, 50]) which brings the computational tools of branch and bound algorithms together with the theoretical tools of polyhedral combinatorics. In 1998, Applegate, Bixby, Chvatal, and Cook used this technique to solve a Traveling Salesman Problem instance with 13,509 cities, a full order of magnitude ....

.... Most DOPs are in the complexity class NP complete, so there is little hope of finding provably e#cient algorithms [40] Nevertheless, intelligent search algorithms, such as LP based branch and bound (to be described below) have been tremendously successful at tackling these di#cult problems [50]. 4.1 Branch and Bound Branch and bound is the broad class of algorithms from which branch, cut, and price has evolved. A branch and bound algorithm uses a divide and conquer strategy to partition the solution space into subproblems and then optimizes individually over each subproblem. For ....

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Ho#man, K., and Padberg, M.: LP-Based Combinatorial Problem Solving. Annals of Operations Research 4, 145, 1985/6

....constraint solving but also in constrained optimization we need a solved form that supports optimization. One of the most powerful techniques to obtain provably good solutions of combinatorial optimization problems is linear optimization combined with the generation of polyhedral cutting planes [HP85] A discrete optimization problem over the 0 1 set S 10 0 1 0 1 . ....

K. Hoffman and M. Padberg. LP-based combinatorial problem solving. Annals of Operations Research, 4:145 -- 194, 1985.

....of large scale mixed integer programming (MIP) problems requires formulations whose linear programming (LP) relaxations give a good approximation to the convex hull of feasible solutions. In the last decade, a great deal of attention has been given to the branch and cut approach to solving MIPs. Hoffman and Padberg [1985], and Nemhauser and Wolsey [1988] give general expositions of this methodology. The basic idea of branch and cut is simple. Classes of valid inequalities, preferably facets of the convex hull of feasible solutions, are left out of the LP relaxation because there are too many constraints to handle ....

K. Hoffman and M. Padberg (1985). LP-based combinatorial problem solving. Annals of Operations Research 4, 145-194.

....or cuts and do some preprocessing which results in coefficient reduction and fixing or elimination of variables. In the last decade, these techniques which provide a closer linear description of the polytope associated with the MIP, were applied to programs with combinatorial structure [3]. The success of these methods additionally are due to improvements of the simplex algorithm and interior point methods for solving linear programs (LP) Variable Fixing and Eliminating To reduce the size of MIPs, it is often helpful to take a closer look at the problem structure. Sometimes, ....

Hoffman, K.L. and Padberg, M.: LP-Based Combinatorial Problem Solving. Annals of Operations Research, 4:145--194, 1991.

....the right hand side of the linearized version of (2) can be increased to dae e =fl efl . Furthermore, the complete program was processed by CPLEX MIP presolve [CPL95] A special node selection scheme, the use of special ordered sets, and the inclusion of some violated cover inequalities [HP85] in every B B node completes the algorithm. After preprocessing the mentioned example reduces to 1547 variables, 139 constraints, and 18192 nonzeros. The value of the LP relaxation increases from 6920.43 to 7577.53 and gives a better lower bound for the B B algorithm. This example was solved in ....

K. Hoffman and M. Padberg. LP-based combinatorial problem solving. Annals of Operations Research, 4:145--194, 1985.

.... introduced by Grotschel, Junger, and Reinelt [7] and Padberg and Rinaldi [22] is enhanced branch and bound where the LP relaxation is tightened at nodes of the tree by the addition of valid inequalities that are not satisfied by the current solution to the LP relaxation (see Hoffman and Padberg [13] and Nemhauser and Wolsey [20] for expositions) Lifted cover inequalities (LCIs) are valid inequalities derived from a knapsack constraint. A cover inequality simply states that not all of the variables in a set can equal one and lifting strengthens the cover inequality by including in the ....

K. Hoffman and M. Padberg. Lp-based combinatorial problem solving. Annals of Operations Research, 4:145--194, 1985.

....programming (BIP) is the problem maxfcx : Ax d; x 2 B n g ; where A 2 ZZ m Thetan is an integer matrix, d 2 ZZ m is an integer vector, c 2 IR n and B stands for the set f0; 1g. BIP is NP hard. Branch and cut is one of the most successful approaches to BIP ( for details see e.g. [5], 6] 7] 8] An important subroutine is to generate cuts from a single knapsack constraint. Suppose a is a row of the matrix A and let b be the corresponding coordinate of the right hand side d. To generate valid inequalities for BIP, we can generate valid inequalities for the polytope P = ....

K. Hoffman and M.W. Padberg, LP-based Combinatorial Problem Solving, Annals of Operations Research 4, 145-194, 1985.

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K.L. Hoffman and M. Padberg (1985). "LP-based Combinatorial Problem Solving," Annals Operations Research, 4, 145-194.

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K. Hoffman and M. Padberg. LP-based combinatorial problem solving. Annals of Operations Research, 4:145--194, 1985.

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K. Hoffman and M. Padberg, LP-Based Combinatorial Problem Solving, Annals of Operations Research 4 (1985/86), 145.

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K. Hoffman and M. Padberg, LP-Based Combinatorial Problem Solving, Annals of Operations Research 4 (1985/86), 145.

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K.L. Hoffman, M.W. Padberg (1985). LP-based combinatorial problem solving.

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