Exploiting Solving Phases for Mixed-Integer Programs

Hendel, Gregor

doi:10.1007/978-3-319-42902-1_1

Cited by 6 publications

(6 citation statements)

References 6 publications

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“…See also the discussion in [2,3,31]. As shift values we use 10 seconds for averaging over running time and 5% for averaging over gap closed values.…”

Section: Methodsmentioning

confidence: 99%

On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

Müller

Muñoz

Gasse

et al. 2020

Integer Programming and Combinatorial Optimization

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The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global -optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm's ability to generate strong dual bounds through extensive computational experiments.

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“…See also the discussion in [2,3,31]. As shift values we use 10 seconds for averaging over running time and 5% for averaging over gap closed values.…”

Section: Methodsmentioning

confidence: 99%

On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

Müller

Muñoz

Gasse

et al. 2020

Integer Programming and Combinatorial Optimization

View full text Add to dashboard Cite

show abstract

“…This measure avoids results being dominated by outliers with large absolute values, and also avoids an over-representation of differences among very small values. See also the discussion in [1,2,29]. As shift values we use 10 seconds for averaging over running time and 5% for averaging over gap closed values.…”

Section: Stabilization Detailsmentioning

confidence: 99%

On generalized surrogate duality in mixed-integer nonlinear programming

et al. 2021

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The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global $$\epsilon $$ ϵ -optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.

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“…In general, we compute average values for runtime and number of solving nodes as shifted geometric mean which reduces the sensitiviy to outliers, cf. [1,16].…”

Section: Computational Evaluationmentioning

confidence: 99%

Advances in verification of ReLU neural networks

Rössig

Petković

2020

J Glob Optim

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We consider the problem of verifying linear properties of neural networks. Despite their success in many classification and prediction tasks, neural networks may return unexpected results for certain inputs. This is highly problematic with respect to the application of neural networks for safety-critical tasks, e.g. in autonomous driving. We provide an overview of algorithmic approaches that aim to provide formal guarantees on the behaviour of neural networks. Moreover, we present new theoretical results with respect to the approximation of ReLU neural networks. On the other hand, we implement a solver for verification of ReLU neural networks which combines mixed integer programming with specialized branching and approximation techniques. To evaluate its performance, we conduct an extensive computational study. For that we use test instances based on the ACAS Xu system and the MNIST handwritten digit data set. The results indicate that our approach is very competitive with others, i.e. it outperforms the solvers of Bunel et al. (in: Bengio, Wallach, Larochelle, Grauman, Cesa-Bianchi, Garnett (eds) Advances in neural information processing systems (NIPS 2018), 2018) and Reluplex (Katz et al. in: Computer aided verification—29th international conference, CAV 2017, Heidelberg, Germany, July 24–28, 2017, Proceedings, 2017). In comparison to the solvers ReluVal (Wang et al. in: 27th USENIX security symposium (USENIX Security 18), USENIX Association, Baltimore, 2018a) and Neurify (Wang et al. in: 32nd Conference on neural information processing systems (NIPS), Montreal, 2018b), the number of necessary branchings is much smaller. Our solver is publicly available and able to solve the verification problem for instances which do not have independent bounds for each input neuron.

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Exploiting Solving Phases for Mixed-Integer Programs

Cited by 6 publications

References 6 publications

On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

On generalized surrogate duality in mixed-integer nonlinear programming

Advances in verification of ReLU neural networks

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