Key Ideas of Algorithm

The algorithm that we will present, is based on the following key ideas:

Instead of immediately computing candidates for error trajectories in one piece, we com-pute trajectories that are only intended to be pieces of the final error trajectory (we will call themtrajectory segmentsin the following). By forming sequences of such trajectory segments we will get candidates for error trajectories.

We reformulate the hybrid systems falsification problem into an optimization problem using an objective function that measures the distance of a sequence of trajectory seg-ments to being an error trajectory.

We allow various strategies for computing the trajectory segments (e.g., random global search). In order to handle the lack of a natural distance measure that can be used for local optimization over the discrete modes we connect the discrete modes by trajectory segments.

136 Jan Kuˇrátko and Stefan Ratschan We use continuous local search to minimize the distance of a sequence of trajectory seg-ments to being an error trajectory. This glues together the trajectory segseg-ments and moves the first and last point toward the set of initial and unsafe states. Note that here we do not only move the trajectory segments, but we also optimize their length.

We initialize the set of computed trajectory segments in such a way that we can use continuous local search from the beginning.

The algorithm can be instantiated with a variety of heuristics and strategies. In the talk we will present some variants of the algorithm and the results of computational experiments with them. We will also discuss conditions under which we succeeded to prove global convergence of the algorithm.

4. Conclusion

We use techniques from global optimization to solve the problem of falsification of hybrid dynamical systems. In general, the application of global optimization techniques to hybrid systems promises to be highly fruitful.

References

[1] H. Abbas and G. Fainekos. Linear hybrid system falsification through local search. InAutomated Technology for Verification and Analysis, volume 6996, pages 503–510. Springer, 2011.

[2] Y. Annpureddy, C. Liu, G. Fainekos, and S. Sankaranarayanan. S-TaLiRo: A tool for temporal logic falsification for hybrid systems. In Parosh Aziz Abdulla and K.Rustan M. Leino, editors,Tools and Algorithms for the Construction and Analysis of Systems, volume 6605 ofLecture Notes in Computer Science, pages 254–257. Springer Berlin Heidelberg, 2011.

[3] U.M. Ascher and L.R. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1st edition, 1998.

[4] M. Gendreau and J.Y. Potvin, editors.Handbook of Metaheuristics. Springer, 2nd edition, 2010.

[5] M. Locatelli and F. Schoen.Global Optimization–Theory, Algorithms, and Applications. SIAM, 2013.

[6] T. Nghiem, S. Sankaranarayanan, G. Fainekos, F. Ivanˇci´c, Aarti Gupta, and George J. Pappas. Monte-Carlo techniques for falsification of temporal properties of non-linear hybrid systems. InHSCC’10, pages 211–220, New York, NY, USA, 2010. ACM.

[7] L.M. Rios and N.V. Sahinidis. Derivative-free optimization: A review of algorithms and comparison of soft-ware implementations.Journal of Global Optimization, pages 1–47, 2012.

[8] F. Schoen. Two-phase methods for global optimization. In P.M. Pardalos and H.E. Romeijn, editors,Handbook of Global Optimization, volume 62 ofNonconvex Optimization and Its Applications, pages 151–177. Springer US, 2002.

[9] A. Zutshi, S. Sankaranarayanan, J.V. Deshmukh, and J. Kapinski. A trajectory splicing approach to concretiz-ing counterexamples for hybrid systems. InCDC’13, 2013.

Proceedings of MAGO 2014, pp. 137 – 140.

MINLPLib 2

Stefan Vigerske

GAMS Development Corp., svigerske@gams.com

Abstract Since 2001, the Mixed-Integer Nonlinear Programming Library (MINLPLib)¹[1] and the GLOBAL Library (GLOBALLib)² have provided algorithm developers with a varied set of both theoretical and practical (MI)NLP test models. In this presentation, we report on recent progress on extending, updating, and categorizing MINLPLib and GLOBALLib. We hope that the updated library can be a starting point to define a widely accepted test set to evaluate the performance of NLP and MINLP solving software.

Keywords: mixed-integer nonlinear programming, nonlinear programming, global optimization, instance li-brary, benchmarking

1. Introduction

Collection of instantiations of mathematical programming models play an important rule for solver software developers. The task of such collections is to provide access to a wide set of interesting problem instances with different characteristics. Especially commercial solver vendors test their solver on thousands of test problems before releasing a new software ver-sion. Additionally, the evaluation of algorithmic improvements (in terms of robustness and efficiency) requires well-balanced test sets of significantly many real-world instances.

The Netlib collection³[3] had an important impact on the field of Linear Programming. Still today, developers of LP solvers test and compare their implementations on this collection. By allowing for independent comparisons of Linear Programming solvers, this collection con-tributed to make Linear Programming solvers as reliable and efficient at they are today. Later, the MIPLIB collection⁴with its regular updates [5] has become the standard test set to compare the performance of Mixed-Integer Linear Programming solvers.

In the area of Global Optimization for (Mixed-Integer) Nonlinear Programming, several collections of model instances have been made available in various formats. In 2001, the MINLPLib collection was released [1], which integrated instances from the GAMS Model Li-brary⁵, MacMINLP⁶, the MINOPT library⁷, and [2]⁸in a single collection using one common format (GAMS). Additionally, the CONVERT tool to translate GAMS models into other for-mats, including AMPL, BARON, and LINGO, was created. Initially, MINLPLib consisted of 136 instances, most of them originating from different applications. The size of the instances varied from tiny (e.g., 1 equation and 5 variables) to huge (e.g., 24972 equations and 23827 variables of which 10920 are binary). Over the years, additional instances were contributed from various sources, so that the MINLPLib consisted of≈270instances by the beginning of 2013. Similarly, the GLOBALLib collection of nonlinear programming (NLP) instances was

1http://www.gamsworld.org/minlp/minlplib.htm

2http://www.gamsworld.org/global/globallib.htm

3http://www.netlib.org/lp/index.html

4http://miplib.zib.de/

5http://www.gams.com/modlib/modlib.htm

6http://www-unix.mcs.anl.gov/ leyffer/MacMINLP/

7http://titan.princeton.edu/MINOPT/modlib/Tests/

8http://titan.princeton.edu/TestProblems/

138 Stefan Vigerske

MINOPT Model Library Vecchietti library Floudas e.a. handbook GAMS Model Library GAMS clients

Westerlund

MacMINLP

BARON book other MINLPLib 1 instance sources (268 in total)

GloMIQO test library MINOPT Model Library Vecchietti library Floudas e.a. handbook GAMS Model Library GAMS clients Westerlund MacMINLP BARON book

minlp.org

other

CMU-IBM MINLP

POLIP

MINLPLib 2 instance sources (817 in total)

Figure 1: Source of Instances in MINLPLib at the beginning of 2013 (left) and now (right).

released in 2001. It originally consisted of 256 instances from [2] and the GAMS Model library.

Today, it comprises 392 instances.

Even though MINLPLib and GLOBALLib have been extremely useful as test sets for solver developers, the inclusion of many very easy instances, many very hard instances, and large homogeneous test sets makes these collections unsuited as a benchmark set to compare global or local solvers. On the other hand, the success of Netlib and the MIPLIB collections raises the hope that a commonly accepted benchmark set of NLP and MINLP instances could enor-mously contribute to the development of efficient and reliable global NLP and MINLP solvers.

As a first step towards this direction, we have started in 2013 with a renovation of the MINLPLib infrastructure and instance collection. In the following, we highlight some of these developments.