MPC

MPC 2010, ISSUE 3-4



Mathematical Programming Computation, Volume 2, Issue 3-4, December 2010

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

Martin S. Andersen, Joachim Dahl, Lieven Vandenberghe

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones.We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.

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Mathematical Programming Computation, Volume 2, Issue 3-4, December 2010

Alternating direction augmented Lagrangian methods for semidefinite programming

Zaiwen Wen, Donald Goldfarb, Wotao Yin

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.

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Mathematical Programming Computation, Volume 2, Issue 3-4, December 2010

A heuristic to generate rank-1 GMI cuts

Sanjeeb Dash, Marcos Goycoolea

Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts.

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Mathematical Programming Computation, Volume 2, Issue 3-4, December 2010

Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems

Artur Pessoa, Eduardo Uchoa, Marcus Poggi de Aragão, Rosiane Rodrigues

This paper presents an exact algorithm for the identical parallel machine scheduling problem over a formulation where each variable is indexed by a pair of jobs and a completion time.We show that such a formulation can be handled, in spite of its huge number of variables, through a branch cut and price algorithm enhanced by a number of practical techniques, including a dynamic programming procedure to fix variables by Lagrangean bounds and dual stabilization. The resulting method permits the solution of many instances of the P||\sum{wj Tj} problem with up to 100 jobs, and having 2 or 4 machines. This is the first time that medium-sized instances of the P||\sum{wj Tj} have been solved to optimality.

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Mathematical Programming Computation, Volume 2, Issue 3-4, December 2010

An inexact interior point method for L1-regularized sparse covariance selection

Lu Li, Kim-Chuan Toh

Sparse covariance selection problems can be formulated as logdeterminant (log-det) semidefinite programming (SDP) problems with large numbers of linear constraints. Standard primal–dual interior-point methods that are based on solving the Schur complement equation would encounter severe computational bottlenecks if they are applied to solve these SDPs. In this paper, we consider a customized inexact primal–dual path-following interior-point algorithm for solving large scale log-det SDP problems arising from sparse covariance selection problems. Our inexact algorithm solves the large and ill-conditioned linear system of equations in each iteration by a preconditioned iterative solver. By exploiting the structures in sparse covariance selection problems, we are able to design highly effective preconditioners to efficiently solve the large and ill-conditioned linear systems. Numerical experiments on both synthetic and real covariance selection problems show that our algorithm is highly efficient and outperforms other existing algorithms.

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MPC 2010, ISSUE 2



Mathematical Programming Computation, Volume 2, Issue 2, June 2010

A library of local search heuristics for the vehicle routing problem

Chris Groër, Bruce Golden, Edward Wasil

The vehicle routing problem (VRP) is a difficult and well-studied combinatorial optimization problem. Real-world instances of the VRP can contain hundreds and even thousands of customer locations and can involve many complicating constraints, necessitating the use of heuristic methods. We present a software library of local search heuristics that allows one to quickly generate solutions to VRP instances. The code has a logical, object-oriented design and uses efficient data structures to store and modify solutions. The core of the library is the implementation of seven local search operators that share a similar interface and are designed to be extended to handle additional options with minimal code change. The code is well-documented, straightforward to compile, and is freely available online. The code contains several applications that can be used to generate solutions to the capacitated VRP. Computational results indicate that these applications are able to generate solutions that are within about one percent of the best-known solution on benchmark problems.

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Mathematical Programming Computation, Volume 2, Issue 2, June 2010

Efficient high-precision matrix algebra on parallel architectures for nonlinear combinatorial optimization

John Gunnels, Jon Lee, Susan Margulies

We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives.

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Mathematical Programming Computation, Volume 2, Issue 2, June 2010

The MCF-separator: detecting and exploiting multi-commodity flow structures in MIPs

Tobias Achterberg, Christian Raack

Given a general mixed integer program, we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of Scip and Cplex. We make use of the complemented mixed integer rounding framework but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speeds-up the computation for a large set of mixed integer programs coming from network design problems by a factor two on average.We show that almost 10% of the instances in general testsets contain consistent embedded networks. For these instances the computation time is decreased by 18% on average.

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MPC 2010, ISSUE 1



Mathematical Programming Computation, Volume 2, Issue 1, March 2010

Optimizing a polyhedral-semidefinite relaxation of completely positive programs

Samuel Burer

It has recently been shown (Burer, Math Program 120:479–495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are NP-hard in general. A basic tractable relaxation is gotten by approximating the completely positive matrices with doubly nonnegative matrices, i.e., matrices which are both nonnegative and positive semidefinite, resulting in a doubly nonnegative program (DNP). Optimizing a DNP, while polynomial, is expensive in practice for interior-point methods. In this paper, we propose a practically efficient decomposition technique, which approximately solves the DNPswhile simultaneously producing lower bounds on the original NQP. We illustrate the effectiveness of our approach for solving the basic relaxation of box-constrained NQPs (BoxQPs) and the quadratic assignment problem. For one quadratic assignment instance, a best-known lower bound is obtained. We also incorporate the lower bounds within a branch-andbound scheme for solving BoxQPs and the quadratic multiple knapsack problem. In particular, to the best of our knowledge, the resulting algorithm for globally solving BoxQPs is the most efficient to date.

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Mathematical Programming Computation, Volume 2, Issue 1, March 2010

On solving trust-region and other regularised subproblems in optimization

Nicholas I. M. Gould, Daniel P. Robinson, H. Sue Thorne

The solution of trust-region and regularisation subproblems that arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Moré and Sorensen, methods that obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems.

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Mathematical Programming Computation, Volume 2, Issue 1, March 2010

A new relaxation framework for quadratic assignment problems based on matrix splitting

Jiming Peng, Hans Mittelmann, Xiaoxue Li

Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the so-called symmetric mappings that can be used to derive strong cuts for the proposed relaxation model. We show that the bounds based on the new models are comparable to some strong bounds in the literature. Promising experimental results based on the new relaxations are reported.

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