While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sp...
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While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These are also applicable for semidefinite relaxations via the spectral bundle method, which allows to exploit structural properties like sparsity. In order to evaluate the relative strengths of linear and semidefinite approaches for large sparse instances, we set up a common branch-and-cut framework for linear and semidefinite relaxations of the minimum graph bisection problem. It incorporates separation algorithms for valid inequalities of the bisection cut polytope described in a recent study by the authors. While the problem specific cuts help to strengthen the linear relaxation significantly, the semidefinite bound profits much more from separating the cycle inequalities of the cut polytope on a slightly enlarged support. Extensive numerical experiments show that this semidefinite branch-and-cut approach without problem specific cuts is a superior choice to the classical simplex approach exploiting bisection specific inequalities on a clear majority of our large sparse test instances from VLSI design and numerical optimization.
Semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection. Using the spectral bundle method it is possible to exploit structural properties of the u...
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ISBN:
(纸本)9783540688860
Semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection. Using the spectral bundle method it is possible to exploit structural properties of the underlying problem and to apply, even to sparse large scale instances, cutting plane methods, probably the most successful technique in linear programming. We set up a common branch-and-cut framework for linear and semidefinite relaxations of the minimum graph bisection problem,. It incorporates separation algorithms for valid inequalities presented in the recent study [2] of the facial structure of the associated polytope. Extensive numerical experiments show that the semidefinite branch-and-cut approach outperforms the classical simplex approach on a clear majority of the sparse large scale test instances. On instances from compiler design the simplex approach is faster.
For trunk packing problems only few approximation schemes are known, mostly designed for the European standard DIN 70020 [6] with equally sized boxes [8, 9, 11, 12]. In this paper two discretized approaches for the US...
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ISBN:
(纸本)9783540728443
For trunk packing problems only few approximation schemes are known, mostly designed for the European standard DIN 70020 [6] with equally sized boxes [8, 9, 11, 12]. In this paper two discretized approaches for the US standard SAE J1100 [10] are presented, which make use of different box sizes. An exact branch-and-bound algorithm for weighted independent sets on graphs is given, using the special structure of the SAE standard. Another branch-and-bound packing algorithm using linear programs is presented. With these algorithms axis-oriented packings of different box sizes in an arbitrary trunk geometry can be computed efficiently.
We discuss an open source implementation and preliminary computational testing of three variants of the Balas-Perregaard procedure for generating lift-and-project cuts from the original simplex tableau, two of which a...
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ISBN:
(纸本)9783540727910
We discuss an open source implementation and preliminary computational testing of three variants of the Balas-Perregaard procedure for generating lift-and-project cuts from the original simplex tableau, two of which are new. Variant 1 is the original procedure of [6] with minor modifications. Variant 2 uses a new procedure for choosing the pivot element: After identifying the set of row candidates for an improving pivot, the pivot element (and column) is chosen by optimizing over the entries of all candidate rows. Finally, Variant 3 replaces the source row with its disjunctive modularization, and after each pivot it again modularizes the resulting source row. We report on computational results with the above three variants and their combinations on 65 MIPLIB.3 instances.
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