TY - GEN A1 - Helmberg, Christoph A1 - Röhl, Stefan T1 - A Case Study of Joint Online Truck Scheduling and Inventory Management for Multiple Warehouses N2 - For a real world problem --- transporting pallets between warehouses in order to guarantee sufficient supply for known and additional stochastic demand --- we propose a solution approach via convex relaxation of an integer programming formulation, suitable for online optimization. The essential new element linking routing and inventory management is a convex piecewise linear cost function that is based on minimizing the expected number of pallets that still need transportation. For speed, the convex relaxation is solved approximately by a bundle approach yielding an online schedule in 5 to 12 minutes for up to 3 warehouses and 40000 articles; in contrast, computation times of state of the art LP-solvers are prohibitive for online application. In extensive numerical experiments on a real world data stream, the approximate solutions exhibit negligible loss in quality; in long term simulations the proposed method reduces the average number of pallets needing transportation due to short term demand to less than half the number observed in the data stream. T3 - ZIB-Report - 05-11 KW - convex relaxation KW - integer programming KW - stochastic demand KW - network models KW - large scale problems KW - bundle method KW - logistics KW - vehicle routing Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8454 ER - TY - GEN A1 - Helmberg, Christoph T1 - Numerical Evaluation of SBmethod N2 - We report numerical results for SBmethod --- a publically available implementation of the spectral bundle method --- applied to the 7$^{th}$ DIMACS challenge test sets that are semidefinite relaxations of combinatorial optimization problems. The performance of the code is heavily influenced by parameters that control bundle update and eigenvalue computation. Unfortunately, no mathematically sound guidelines for setting them are known. Based on our experience with SBmethod, we propose heuristics for dynamically updating the parameters as well as a heuristc for improving the starting point. These are now the default settings of SBmethod Version 1.1. We compare their performance on the DIMACS instances to our previous best choices for Version 1.0. SBmethod Version 1.1 is also part of the independent DIMACS benchmark by H.~Mittelmann. Based on these results we try to analyze strengths and weaknesses of our approach in comparison to other codes for large scale semidefinite programming. T3 - ZIB-Report - 01-37 KW - semidefinite programming KW - large scale methods KW - computational KW - semidefinite relaxations Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6637 ER - TY - GEN A1 - Helmberg, Christoph T1 - A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations N2 - The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 0-1 programming problems by odd-cycle inequalities. We also consider separating odd-cycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for max-cut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably. T3 - ZIB-Report - 01-26 KW - bisection KW - equicut KW - max-cut KW - semidefinite programming KW - spectral bundle method KW - subgradient method KW - quadratic 0-1 programming Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6527 ER - TY - THES A1 - Helmberg, Christoph T1 - Semidefinite Programming for Combinatorial Optimization N2 - This book offers a self-contained introduction to the field of semidefinite programming, its applications in combinatorial optimization, and its computational methods. We equip the reader with the basic results from linear algebra on positive semidefinite matrices and the cone spanned by them. Starting from linear programming, we introduce semidefinite programs and discuss the associated duality theory. We then turn to semidefinite relaxations of combinatorial optimization and illustrate their interrelation. In the second half we deal with computational methods for solving semidefinite programs. First, the interior point approach, its iteration complexity, and implementational issues are discussed. Next, we explain in great detail the spectral bundle method, which is particularly suited for large scale semidefinite programming. One of the most successful techniques in integer linear programming is the cutting plane approach which improves an initial relaxation by adding violated inequalities. We explore possibilities to combine the two solution methods with the cutting plane approach in order to strengthen semidefinite relaxations of combinatorial optimization problems. T3 - ZIB-Report - 00-34 KW - positive semidefinite matrices KW - semidefinite cone KW - semidefinite Programming KW - semidefinite duality KW - combinatorial optimization KW - max-cut KW - quadrati Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6022 ER -