U.-U. Haus, M. Koppe, and R. Weismantel, The Integral Basis Method for Integer Programming, Mathematical Methods of Operations Research 53 (3), 2001  Home/Search   Document Details and Download   Summary   Related Articles   Check

....how to make our algorithms practical. A simple idea would be to project our formulations to the space of the original variables. However, we prefer the use of additional variables indexed by the subset al..gebra as they reveal useful structure of the problem. We note that Haus, Koppe and Weismantel [HKW01] have introduced algorithms for solving general integer programs which rely on explicitly adding new variables, though in a rather di#erent form than our algorithms. Also, we point out the result in Section 3.3 a polynomial enlargement of a formulation can imply an exponential number of facets, ....

U.-U. Haus, M. Koppe and R. Weismantel, The Integral Basis Method for integer programming. Mathematical Methods of Operations Research 3 (2001), 353--361.

.... present paper are so called augmentation algorithms, in which a feasible solution is iteratively improved (augmented) until no further improvement is possible (and it can be proved that this is the case) Some recent results on augmentation algorithms can be found in Firla et al. 8] Haus et al. [14], Schulz et al. 22] Thomas [23] and Urbaniak et al. 24] Recently, Weismantel [25] suggested the possibility of somehow combining elements of augmentation and branch and cut algorithms, to yield an augmentand branch and cut algorithm for integer programming with the convenient acronym ABC. ....

U.-U. Haus, M. Koppe & R. Weismantel (2000) The integral basis method for integer programming. Math. Meth. of Oper. Res. 53, 353-361.

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U.-U. Haus, M. Koppe, and R. Weismantel, The Integral Basis Method for Integer Programming, Mathematical Methods of Operations Research 53 (3), 2001

....to Gomory [5] Balas, Ceria, and Cornujols [2] and Nemhauser and Wolsey [11] for cutting plane algorithms applicable to mixed integer programs involving 0 1 or general integer variables. We propose in this paper a purely primal approach that extends the earlier work of Haus, Kppe, and Weismantel [7, 8] from the pure integer setting to a mixed integer linear optimization problem. More precisely, we assume that a primal feasible solution is at hand. This is then turned into a basic feasible solution of an appropriate tableau. Iteratively we reformulate the tableau by eliminating columns with ....

....programs has recently occurred in the work of Aardal, Hurkens, and Lenstra [1] where lattice basis reduction is employed to reformulate integer programs of a certain class in a way such that the number of branch and bound nodes is reduced. The Integral Basis Method by Haus, Kppe, and Weismantel [7, 8] is a primal integer programming method that constructs a sequence of integer programs that are proper linear reformulations of the original integer program, until an augmentation vector is found or optimality is proved. Likewise, for solving stable set problems, Gentile et al. 4] studied a ....

U.-U. Haus,M.Kppe,and R. Weismantel. The Integral Basis Method for integer programming. Mathematical Methods of Operations Research, 53(3):353--361, 2001.

....Gomory [4] Balas, Ceria, and Cornuejols [2] and Nemhauser and Wolsey [9] for cutting plane algorithms applicable to mixed integer programs involving 0 1 or general integer variables. We propose in this paper a purely primal approach that extends the earlier work of Haus, K oppe, and Weismantel [6, 7] from the pure integer setting to a mixed integer linear optimization problem. More precisely, we assume that a primal feasible solution is at hand. This is then turned into a basic feasible solution of an appropriate tableau. Iteratively we reformulate the tableau by eliminating columns with ....

....programs has recently occurred in the work of Aardal, Hurkens, and Lenstra [1] where lattice basis reduction is employed to reformulate integer programs of a certain class in a way such that the number of branch and bound nodes is reduced. The Integral Basis Method by Haus, K oppe, and Weismantel [6, 7] is a primal integer programming method that constructs a sequence of integer programs that are proper linear reformulations of the original integer program, until an augmentation vector is found or optimality is proved. Likewise, for solving stable set problems, Gentile et al. 3] studied a ....

U.-U. Haus,M.K oppe,and R. Weismantel. The Integral Basis Method for integer programming. Mathematical Methods of Operations Research, 53(3):353--361, 2001.

....FKZ 0037KD0099 and FKZ 2495A 0028G of the Kultusministerium of Sachsen Anhalt. Supported by a Gerhard Hess Preis and grant WE 1462 of the Deutsche Forschungsgemeinschaft, and by the European DONET program TMR ERB FMRX CT98 0202. 1 2 just obtained to start the Integral Basis Method (see [HKW01a, HKW01b] This is an algorithm that, working in a simplex tableau, iteratively substitutes non basic columns by newly generated columns, following the track laid by Balas and Padberg in [BP75] where properties of the simplex tableau were exploited in the case of set partitioning problems. In ....

....assume that all non basic variables are 0 1, whereas the basic variables may be 9 general integer variables. This setting is general enough to deal with 0 1 integer programs, for which computational results are shown in section 5. We work with a variant of the Integral Basis Method presented in [HKW01a] which is based on irreducible solutions to discrete relaxations of the projection F N of the feasible region F onto the non basic variables: Definition 11. a) For a tableau (6) let F N = # z # Z n m : Az # b # . We call a set F # N = # z # Z n m : A # z # b # # a ....

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U.-U. Haus, M. Koppe, and R. Weismantel, The integral basis method for integer programming, Mathematical Methods of Operations Research 53 (2001), no. 3, 353--361.

....Method has been successfully applied to hard 0 1 integer programs from the benchmark library MIPLIB. For implementation notes and computational results the reader is referred to the companion paper [HKW01b] We also remark that a short outline of our algorithm has already appeared in the paper [HKW01a]. The work [BP75] of Balas and Padberg can be regarded as an early predecessor of our method. They exploit properties of the simplex tableau in the case of set partitioning problems, whereas our method works for general integer programs. 2. From Integer Programming to Basic Feasible Solutions We ....

U.-U. Haus, M. Koppe, and R. Weismantel, The Integral Basis Method for integer programming, Mathematical Methods of Operations Research 53 (2001), no. 3, 353--361. Haus, Koppe, Weismantel: A Primal All-Integer Algorithm Based on Irreducible Solutions

....Their basic technique was to replace a column of the current simplex tableau with a set of new columns in order to guarantee the next pivot to lead to an integral basic feasible solution. These ideas were generalized to the case of general integer programming by Haus, Koppe, and Weismantel [HKW01a], who called their method the Integral Basis Method. This method does neither require cutting planes nor enumeration techniques. In each major step the algorithm either returns an augmenting direction that is applicable at the given feasible point and yields a new feasible point with better ....

....to remove the column of the non basic variable x v 1 from the tableau and replace it by certain sums of other columns of the tableau. This technique is closely related to the integral basis method by Haus, Koppe, and Weismantel, which is a primal algorithm for solving general integer programs [HKW01a]. We shall use the notation x v # x w for a new variable associated with a column that is the sum of the columns for x v and x w . D######### 16. For an odd alternating path of cliques ( v 1 , Q 2,3 , Q 4,5 , Q 2l,2l 1 ) we define the corresponding alternating path substitution in ....

U.-U. Haus, M. Koppe, and R. Weismantel, The Integral Basis Method for integer programming, Mathematical Methods of Operations Research 53 (2001), no. 3, 353--361. 23 24

....Balas and Padberg can be regarded as an early predecessor of our method. They exploit properties of the simplex tableau in the case of set partitioning problems, whereas our method works for general integer programs. We remark that a short outline of our algorithm has already appeared in the paper [HKW01a]. For thorough computational results the reader is referred to the companion paper [HKW01b] 2 Discrete Relaxations and Irreducible Lattice Points Throughout this section we deal with integral systems of the form S : z # Z n : Az # b , A # Z mn , b # Z m . 1) ....

U.-U. Haus, M. Koppe, and R. Weismantel, The integral basis method for integer programming, Mathematical Methods of Operations Research 53 (2001), no. 3.

....from zero, pivoting on this cut results in an integer pivot step. This approach is based on the primal simplex algorithm and was first proposed by Ben Israel Charnes [1] Simplified variants were given by Young [19, 20] and Glover [7] The integral basis method of Haus, Koppe Weismantel [14, 16] is a second approach to perform this task. It refrains from adding cuts, but replaces the 2 cutting step by a step in which the columns of the given system are manipulated. Roughly speaking, their algorithm iteratively substitutes one column by columns that correspond to irreducible solutions ....

....the columns of the given system are manipulated. Roughly speaking, their algorithm iteratively substitutes one column by columns that correspond to irreducible solutions of certain linear diophantine inequalities, maintaining the integral basis property of the tableau as in invariant. We refer to [14, 16] for more details on this algorithm. The intention of this paper is to recall this classical and nearly forgotten approach of integer pivoting. We demonstrate, that the idea of associating a basic solution of a simplex tableau with a given integer point and performing integer pivots as long as ....

U.-U. Haus, M. Koppe, and R. Weismantel, The Integral Basis Method for Integer Programming, Mathematical Methods of Operations Research 53 (3), 2001

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U.-U. Haus, M. Koppe & R. Weismantel (2001) The integral basis method for integer programming. Math. Meth. of Oper. Res. 53, 353--361.

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