65K05 Mathematical programming methods [See also 90Cxx]
Refine
Year of publication
Document Type
- ZIB-Report (20)
Language
- English (20)
Has Fulltext
- yes (20)
Is part of the Bibliography
- no (20)
Keywords
Institute
- Mathematical Optimization (15)
- ZIB Allgemein (4)
- Numerical Mathematics (1)
We propose a hybrid approach for solving the resource-constrained project scheduling problem which is an extremely hard to solve combinatorial optimization problem of practical relevance. Jobs have to be scheduled on (renewable) resources subject to precedence constraints such that the resource capacities are never exceeded and the latest completion time of all jobs is minimized. The problem has challenged researchers from different communities, such as integer programming (IP), constraint programming (CP), and satisfiability testing (SAT). Still, there are instances with 60 jobs which have not been solved for many years. The currently best known approach, lazyFD, is a hybrid between CP and SAT techniques. In this paper we propose an even stronger hybridization by integrating all the three areas, IP, CP, and SAT, into a single branch-and-bound scheme. We show that lower bounds from the linear relaxation of the IP formulation and conflict analysis are key ingredients for pruning the search tree. First computational experiments show very promising results. For five instances of the well-known PSPLIB we report an improvement of lower bounds. Our implementation is generic, thus it can be potentially applied to similar problems as well.
Pseudo-Boolean problems lie on the border between satisfiability problems, constraint programming, and integer programming. In particular, nonlinear constraints in pseudo-Boolean optimization can be handled by methods arising in these different fields: One can either linearize them and work on a linear programming relaxation or one can treat them directly by propagation. In this paper, we investigate the individual strengths of these approaches and compare their computational performance. Furthermore, we integrate these techniques into a branch-and-cut-and-propagate framework, resulting in an efficient nonlinear pseudo-Boolean solver.
This extended abstract is about algorithms for controlling elevator systems employing destination hall calls, i.e. the passenger provides his destination floor when calling an elevator. We present the first exact algorithm for controlling a group of elevators and report on simulation results indicating that destination hall call systems outperform conventional systems.
Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field. In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method.
This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques from SAT solving. SCIP is available in source code and free for non-commercial use. We demonstrate the usefulness of CIP on two tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we employ the CIP framework to solve chip design verification problems, which involve some highly non-linear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the non-linear constraints by employing constraint programming techniques.
Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or $\frac{1}{2}$, such cuts are known as $\{0,\frac{1}{2}\}$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of $\{0,\frac{1}{2}\}$-cuts is NP-hard. In this paper, we study ways to separate $\{0,\frac{1}{2}\}$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated $\{0,\frac{1}{2}\}$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.
Relaying is a protocol extension for cellular wireless computer networks; in order to utilize radio resources more efficiently, several hops are allowed within one cell. This paper investigates the principle potential of relaying by casting transmission scheduling as a mathematical optimization problem, namely, a linear program. We analyze the throughput gains showing that, irrespective of the concrete scheduling algorithm, performance gains of up to 30\% on average for concrete example networks are achievable.
Relaying -- allowing multiple wireless hops -- is a protocol extension for cellular networks conceived to improve data throughput. Its benefits have only been quantified for small example networks. For assessing its general potential, we define a complex resource allocation\slash{}scheduling problem. Several mathematical models are presented for this problem; while a time-expanded MIP approach turns out intractable, a sophisticated column generation scheme leads to good computational results. We thereby show that for selected cases relaying can increase data throughput by 30\% on the average.
The paper provides a detailed analysis of a short step interior point algorithm applied to linear control constrained optimal control problems. Using an affine invariant local norm and an inexact Newton corrector, the well-known convergence results from finite dimensional linear programming can be extended to the infinite dimensional setting of optimal control. The present work complements a recent paper of Weiser and Deuflhard, where convergence rates have not been derived. The choice of free parameters, i.e. the corrector accuracy and the number of corrector steps, is discussed.
A path following algorithm for linear complementarity problems is presented. Given a point $z$ that approximates a point $z(\tau)$ on the central path with complementarity gap $\tau$, one determines a parameter $\theta\in (0,1)$ so that this point satisfies the hypothesis of the affine invariant Kantorovich Theorem for the equation defining $z((1-\theta)\tau)$. It is shown that $\theta$ is bounded below by a multiple of $n^{-1/2}$, where $n$ is the dimension of the problem. Since the hypothesis of of the Kantorovich Theorem is satisfied the sequence generated by Newton's method, or by the simplified Newton method, will converge to $z((1-\theta)\tau)$. We show that the number of steps required to obtain an acceptable approximation of $z((1-\theta)\tau)$ is bounded above by a number independent of $n$. Therefore the algorithm has $O(\sqrt{n}L)$-iteration complexity. The parameters of the algorithm can be determined in such a way that only one Newton step is needed each time the complementarity gap is decreased.