MPC

MPC 2021, ISSUE 3



Mathematical Programming Computation, Volume 13, Issue 3, September 2021

MIPLIB 2017: data-driven compilation of the 6th mixed-integer programming library

Ambros Gleixner, Gregor Hendel, Gerald Gamrath, Tobias Achterberg, Michael Bastubbe, Timo Berthold, Philipp Christophel, Kati Jarck, Thorsten Koch, Jeff Linderoth, Marco Lübbecke, Hans D. Mittelmann, Derya Ozyurt, Ted K. Ralphs, Domenico Salvagnin, Yuji Shinano

We report on the selection process leading to the sixth version of the Mixed Integer Programming Library, MIPLIB 2017. Selected from an initial pool of 5721 instances, the new MIPLIB 2017 collection consists of 1065 instances. A subset of 240 instances was specially selected for benchmarking solver performance. For the first time, these sets were compiled using a data-driven selection process supported by the solution of a sequence of mixed integer optimization problems, which encode requirements on diversity and balancedness with respect to instance features and performance data.

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Mathematical Programming Computation, Volume 13, Issue 3, September 2021

A triangulation and fill-reducing initialization procedure for the simplex algorithm

Nikolaos Ploskas, Nikolaos V. Sahinidis, Nikolaos Samaras

The computation of an initial basis is of great importance for simplex algorithms since it determines to a large extent the number of iterations and the computational effort needed to solve linear programs. We propose three algorithms that aim to construct an initial basis that is sparse and will reduce the fill-in and computational effort during LU factorization and updates that are utilized in modern simplex implementations. The algorithms rely on triangulation and fill-reducing ordering techniques that are invoked prior to LU factorization. We compare the performance of the CPLEX 12.6.1 primal and dual simplex algorithms using the proposed starting bases against CPLEX using its default crash procedure over a set of 95 large benchmarks (NETLIB, Kennington, Mészáros, Mittelmann). The best proposed algorithm utilizes METIS (Karypis and Kumar in SIAM J Sci Comput 20:359–392, 1998), produces remarkably sparse starting bases, and results in 5% reduction of the geometric mean of the execution time of CPLEX’s primal simplex algorithm. Although the proposed algorithm improves CPLEX’s primal simplex algorithm across all problem types studied in this paper, it performs better on hard problems, i.e., the instances for which the CPLEX default requires over 1000 s. For these problems, the proposed algorithm results in 37% reduction of the geometric mean of the execution time of CPLEX’s primal simplex algorithm. The proposed algorithm also reduces the execution time of CPLEX’s dual simplex on hard instances by 10%. For the instances that are most difficult for CPLEX, and for which CPLEX experiences numerical difficulties as it approaches the optimal solution, the best proposed algorithm speeds up CPLEX by more than 10 times. Finally, the proposed algorithms lead to a natural way to parallelize CPLEX with speedups over CPLEX’s dual simplex of 1.2 and 1.3 on two and four cores, respectively.

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Mathematical Programming Computation, Volume 13, Issue 3, September 2021

Design and implementation of a modular interior-point solver for linear optimization

Mathieu Tanneau, Miguel F. Anjos, Andrea Lodi

This paper introduces the algorithmic design and implementation of Tulip, an open-source interior-point solver for linear optimization. It implements a regularized homogeneous interior-point algorithm with multiple centrality corrections, and therefore handles unbounded and infeasible problems. The solver is written in Julia, thus allowing for a flexible and efficient implementation: Tulip’s algorithmic framework is fully disentangled from linear algebra implementations and from a model’s arithmetic. In particular, this allows to seamlessly integrate specialized routines for structured problems. Extensive computational results are reported. We find that Tulip is competitive with open-source interior-point solvers on the H. Mittelmann’s benchmark of barrier linear programming solvers. Furthermore, we design specialized linear algebra routines for structured master problems in the context of Dantzig–Wolfe decomposition. These routines yield a tenfold speedup on large and dense instances that arise in power systems operation and two-stage stochastic programming, thereby outperforming state-of-the-art commercial interior point method solvers. Finally, we illustrate Tulip’s ability to use different levels of arithmetic precision by solving problems in extended precision.

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Mathematical Programming Computation, Volume 13, Issue 3, September 2021

Deterministic global optimization with Gaussian processes embedded

Artur M. Schweidtmann, Dominik Bongartz, Daniel Grothe, Tim Kerkenhoff, Xiaopeng Lin, Jaromił Najman, Alexander Mitsos

Gaussian processes (Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the free variables and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the “MeLOn—Machine Learning Models for Optimization” toolbox (https://git.rwth-aachen.de/avt.svt/public/MeLOn).

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Mathematical Programming Computation, Volume 13, Issue 3, September 2021

An inexact proximal augmented Lagrangian framework with arbitrary linearly convergent inner solver for composite convex optimization

Fei Li, Zheng Qu

We propose an inexact proximal augmented Lagrangian framework with explicit inner problem termination rule for composite convex optimization problems. We consider arbitrary linearly convergent inner solver including in particular stochastic algorithms, making the resulting framework more scalable facing the ever-increasing problem dimension. Each subproblem is solved inexactly with an explicit and self-adaptive stopping criterion, without requiring to set an a priori target accuracy. When the primal and dual domain are bounded, our method achieves O(1/ϵ√) and O(1/ϵ) complexity bound in terms of number of inner solver iterations, respectively for the strongly convex and non-strongly convex case. Without the boundedness assumption, only logarithm terms need to be added and the above two complexity bounds increase respectively to O~(1/ϵ√) and O~(1/ϵ), which hold both for obtaining ϵ-optimal and ϵ-KKT solution. Within the general framework that we propose, we also obtain O~(1/ϵ) and O~(1/ϵ2) complexity bounds under relative smoothness assumption on the differentiable component of the objective function. We show through theoretical analysis as well as numerical experiments the computational speedup possibly achieved by the use of randomized inner solvers for large-scale problems.

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