MPC

MPC 2015, ISSUE 4



Mathematical Programming Computation, Volume 7, Issue 4, December 2015

Progress in presolving for mixed integer programming

Gerald Gamrath, Thorsten Koch, Alexander Martin, Matthias Miltenberger, Dieter Weninger

This paper describes three presolving techniques for solving mixed integer programming problems (MIPs) that were implemented in the academic MIP solver SCIP. The task of presolving is to reduce the problem size and strengthen the formulation, mainly by eliminating redundant information and exploiting problem structures. The first method fixes continuous singleton columns and extends results known from duality fixing. The second analyzes and exploits pairwise dominance relations between variables, whereas the third detects isolated subproblems and solves them independently. The performance of the presented techniques is demonstrated on two MIP test sets. One contains all benchmark instances from the last three MIPLIB versions, while the other consists of real-world supply chain management problems. The computational results show that the combination of all three presolving techniques almost halves the solving time for the considered supply chain management problems. For the MIPLIB instances we obtain a speedup of 20% on affected instances while not degrading the performance on the remaining problems.

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Mathematical Programming Computation, Volume 7, Issue 4, December 2015

A quasi-Newton algorithm for nonconvex, nonsmooth optimization with global convergence guarantees

Xiaocun Que, Frank E. Curtis

A line search algorithm for minimizing nonconvex and/or nonsmooth objective functions is presented. The algorithm is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed because it typically yields fast convergence to the vicinity of a stationary point, and together with the adaptive GS strategy the algorithm ensures that convergence will continue to such a point. Under suitable assumptions, it is proved that the algorithm converges globally with probability one. The algorithm has been implemented inC++and the results of numerical experiments illustrate the efficacy of the proposed approach.

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Mathematical Programming Computation, Volume 7, Issue 4, December 2015

PEBBL: an object-oriented framework for scalable parallel branch and bound

Jonathan Eckstein, William E. Hart, Cynthia A. Phillips

Parallel Enumeration and Branch-and-Bound Library (PEBBL) is a C++ class library implementing the underlying operations needed to support a wide variety of branch-and-bound algorithms on MPI-based message-passing distributed-memory parallel computing environments. PEBBL can be customized to support applicationspecific operations, while managing the generic aspects of branch and bound, such as maintaining the active subproblem pool across multiple processors, load balancing, and termination detection. PEBBL is designed to provide highly scalable performance on large numbers of processor cores.We describe the basics of PEBBL’s architecture, with emphasis on the features most critical to is high scalability, including its flexible two-level load balancing architecture and its support for a synchronously parallel ramp-up phase.We also present an example application: themaximummonomial agreement problem arising from certain machine learning applications. For sufficiently difficult problem instances, we show essentially linear speedup on over 6000 processor cores, demonstrating a new state of the art in scalability for branch-and-bound implementations. We also show how processor cache effects can lead to reproducibly superlinear speedups.

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Mathematical Programming Computation, Volume 7, Issue 4, December 2015

Computational study of decomposition algorithms for mean-risk stochastic linear programs

Lewis Ntaimo, Tanisha G. Cotton

Mean-risk stochastic programs include a risk measure in the objective to model risk averseness for many problems in science and engineering. This paper reports a computational study of mean-risk two-stage stochastic linear programs with recourse based on absolute semideviation (ASD) and quantile deviation (QDEV). The studywas aimed at performing an empirical investigation of decomposition algorithms for stochastic programs with quantile and deviation mean-risk measures; analyzing how the instance solutions vary across different levels of risk; and understanding when it is appropriate to use a given mean-risk measure. Aggregated optimality cut and separate cut subgradient-based algorithms were implemented for each mean-risk model. Both types of algorithms show similar computational performance for ASD whereas the separate cut algorithm outperforms the aggregated cut algorithm for QDEV. The study provides several insights. For example, the results reveal that the risk-neutral approach is still appropriate for most of the standard stochastic programming test instances due to their uniform or normal-like marginal distributions. However, when the distributions are modified, the risk-neutral approach may no longer be appropriate and the risk-averse approach becomes necessary. The results also show that ASD is a more conservative mean-risk measure than QDEV.

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