{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T06:44:46Z","timestamp":1773384286285,"version":"3.50.1"},"publisher-location":"Cham","reference-count":31,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783031067723","type":"print"},{"value":"9783031067730","type":"electronic"}],"license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022]]},"DOI":"10.1007\/978-3-031-06773-0_16","type":"book-chapter","created":{"date-parts":[[2022,5,19]],"date-time":"2022-05-19T11:24:44Z","timestamp":1652959484000},"page":"299-317","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["ZoPE: A Fast Optimizer for\u00a0ReLU Networks with\u00a0Low-Dimensional Inputs"],"prefix":"10.1007","author":[{"given":"Christopher A.","family":"Strong","sequence":"first","affiliation":[]},{"given":"Sydney M.","family":"Katz","sequence":"additional","affiliation":[]},{"given":"Anthony L.","family":"Corso","sequence":"additional","affiliation":[]},{"given":"Mykel J.","family":"Kochenderfer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,20]]},"reference":[{"key":"16_CR1","unstructured":"Bojarski, M., et al.: End to end learning for self-driving cars, Technical Report (2016). http:\/\/arxiv.org\/abs\/1604.07316"},{"issue":"3","key":"16_CR2","doi-asserted-by":"publisher","first-page":"598","DOI":"10.2514\/1.G003724","volume":"42","author":"KD Julian","year":"2019","unstructured":"Julian, K.D., Kochenderfer, M.J., Owen, M.P.: Deep neural network compression for aircraft collision avoidance systems. AIAA J. Guid. Control Dyn. 42(3), 598\u2013608 (2019)","journal-title":"AIAA J. Guid. Control Dyn."},{"key":"16_CR3","doi-asserted-by":"crossref","unstructured":"Liu, C., Arnon, T., Lazarus, C., Strong, C., Barrett, C., Kochenderfer, M.J.: Algorithms for verifying deep neural networks. Found. Trends\u00ae Optim. 4(3\u20134), 244\u2013404 (2021)","DOI":"10.1561\/2400000035"},{"key":"16_CR4","doi-asserted-by":"crossref","unstructured":"Katz, S.M., Corso, A.L., Strong, C.A., Kochenderfer, M.J.: Verification of image-based neural network controllers using generative models. In: Digital Avionics Systems Conference (DASC) (2021)","DOI":"10.1109\/DASC52595.2021.9594360"},{"key":"16_CR5","doi-asserted-by":"crossref","unstructured":"Julian, K.D., Lee, R., Kochenderfer, M.J.: Validation of image-based neural network controllers through adaptive stress testing (2020)","DOI":"10.1109\/ITSC45102.2020.9294549"},{"key":"16_CR6","doi-asserted-by":"publisher","unstructured":"Katz, S.M., Julian, K.D., Strong, C.A., Kochenderfer, M.J.: Generating probabilistic safety guarantees for neural network controllers. Mach. Learn. , 1\u201329 (2021). https:\/\/doi.org\/10.1007\/s10994-021-06065-9","DOI":"10.1007\/s10994-021-06065-9"},{"key":"16_CR7","unstructured":"Kynk\u00e4\u00e4nniemi, T., Karras, T., Laine, S., Lehtinen, J., Aila, T.: Improved precision and recall metric for assessing generative models. In: Advances in Neural Information Processing Systems (NeurIPS) (2019)"},{"key":"16_CR8","doi-asserted-by":"crossref","unstructured":"Mirman, M., Gehr, T., Vechev, M.: Robustness certification with generative models. In: ACM SIGPLAN International Conference on Programming Language Design and Implementation (2021)","DOI":"10.1145\/3410308"},{"key":"16_CR9","doi-asserted-by":"publisher","unstructured":"Strong, C.A., et al.: Global optimization of objective functions represented by ReLU networks. Mach. Learn. 2010.03258 (2021). https:\/\/doi.org\/10.1007\/s10994-021-06050-2","DOI":"10.1007\/s10994-021-06050-2"},{"key":"16_CR10","unstructured":"Singh, G., Gehr, T., P\u00fcschel, M., Vechev, M.T.: Boosting robustness certification of neural networks. In: International Conference on Learning Representations (2019)"},{"key":"16_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/978-3-030-53288-8_4","volume-title":"Computer Aided Verification","author":"S Bak","year":"2020","unstructured":"Bak, S., Tran, H.-D., Hobbs, K., Johnson, T.T.: Improved geometric path enumeration for verifying ReLU neural networks. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12224, pp. 66\u201396. Springer, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-53288-8_4"},{"key":"16_CR12","doi-asserted-by":"publisher","unstructured":"Katz, G., et al.: The Marabou framework for verification and analysis of deep neural networks. In: Dillig, I., Tasiran, S. (eds.) CAV 2019. LNCS, vol. 11561, pp. 443\u2013452. Springer, Cham (2019). https:\/\/doi.org\/10.1007\/978-3-030-25540-4_26","DOI":"10.1007\/978-3-030-25540-4_26"},{"issue":"2020","key":"16_CR13","first-page":"1","volume":"21","author":"R Bunel","year":"2020","unstructured":"Bunel, R., Mudigonda, P., Turkaslan, I., Torr, P., Lu, J., Kohli, P.: Branch and bound for piecewise linear neural network verification. J. Mach. Learn. Res. 21(2020), 1\u201339 (2020)","journal-title":"J. Mach. Learn. Res."},{"key":"16_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/978-3-319-68167-2_19","volume-title":"Automated Technology for Verification and Analysis","author":"R Ehlers","year":"2017","unstructured":"Ehlers, R.: Formal verification of piece-wise linear feed-forward neural networks. In: D\u2019Souza, D., Narayan Kumar, K. (eds.) ATVA 2017. LNCS, vol. 10482, pp. 269\u2013286. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-68167-2_19"},{"key":"16_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/978-3-319-63387-9_5","volume-title":"Computer Aided Verification","author":"G Katz","year":"2017","unstructured":"Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.: Reluplex: an efficient SMT solver for verifying deep neural networks. In: Majumdar, R., Kun\u010dak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 97\u2013117. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-63387-9_5"},{"key":"16_CR16","unstructured":"Wang, S., Pei, K., Whitehouse, J., Yang, J., Jana, S.: Formal security analysis of neural networks using symbolic intervals. In: USENIX Security Symposium 2018, pp. 1599\u20131614 (2018)"},{"key":"16_CR17","unstructured":"Wu, H., et al.: Parallelization techniques for verifying neural networks. CoRR, vol. abs\/2004.08440 (2020). arXiv: 2004.08440"},{"key":"16_CR18","unstructured":"Tjeng, V., Xiao, K., Tedrake, R.: Evaluating robustness of neural networks with mixed integer programming. In: International Conference on Learning Representations (2017)"},{"key":"16_CR19","unstructured":"Singh, G., Gehr, T., Mirman, M., P\u00fcschel, M., Vechev, M.T.: Fast and effective robustness certification. In: Advances in Neural Information Processing Systems (NeurIPS) (2018)"},{"key":"16_CR20","doi-asserted-by":"crossref","unstructured":"Singh, G., Gehr, T., P\u00fcschel, M., Vechev, M.: An abstract domain for certifying neural networks. In: Proceedings of the ACM on Programming Languages, vol. 3, no. POPL, pp. 1\u201330 (2019)","DOI":"10.1145\/3290354"},{"key":"16_CR21","unstructured":"Fujishige, S.: Submodular Functions and Optimization. Elsevier (2005)"},{"issue":"1","key":"16_CR22","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s00453-018-0436-3","volume":"81","author":"T Kitahara","year":"2019","unstructured":"Kitahara, T., Sukegawa, N.: A simple projection algorithm for linear programming problems. Algorithmica 81(1), 167\u2013178 (2019)","journal-title":"Algorithmica"},{"issue":"4","key":"16_CR23","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1287\/opre.14.4.699","volume":"14","author":"EL Lawler","year":"1966","unstructured":"Lawler, E.L., Wood, D.E.: Branch-and-bound methods: a survey. Oper. Res. 14(4), 699\u2013719 (1966)","journal-title":"Oper. Res."},{"key":"16_CR24","unstructured":"Kochenderfer, M.J., Wheeler, T.A.: Algorithms for Optimization. MIT Press, Cambridge (2019)"},{"key":"16_CR25","doi-asserted-by":"crossref","unstructured":"Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: IEEE Conference on Decision and Control (CDC), pp. 4042\u20134048 (2008)","DOI":"10.1109\/CDC.2008.4738704"},{"key":"16_CR26","unstructured":"Rubies-Royo, V., Calandra, R., Stipanovic, D.M., Tomlin, C.: Fast neural network verification via shadow prices. arXiv preprint arXiv:1902.07247 (2019)"},{"issue":"9","key":"16_CR27","doi-asserted-by":"publisher","first-page":"2805","DOI":"10.1109\/TNNLS.2018.2886017","volume":"30","author":"X Yuan","year":"2019","unstructured":"Yuan, X., He, P., Zhu, Q., Li, X.: Adversarial examples: attacks and defenses for deep learning. IEEE Trans. Neural Netw. Learn. Syst. 30(9), 2805\u20132824 (2019)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"16_CR28","doi-asserted-by":"crossref","unstructured":"Althoff, M., Frehse, G.: Combining zonotopes and support functions for efficient reachability analysis of linear systems. In: IEEE Conference on Decision and Control (CDC), pp. 7439\u20137446 (2016)","DOI":"10.1109\/CDC.2016.7799418"},{"key":"16_CR29","unstructured":"Althoff, M.: On computing the Minkowski difference of zonotopes. arXiv preprint arXiv:1512.02794 (2015)"},{"key":"16_CR30","unstructured":"Lomuscio, A., Maganti, L.: An approach to reachability analysis for feedforward ReLU neural networks. arXiv preprint arXiv:1706.07351 (2017)"},{"key":"16_CR31","unstructured":"Singh, G., Ganvir, R., P\u00fcschel, M., Vechev, M.: Beyond the single neuron convex barrier for neural network certification. In: Advances in Neural Information Processing Systems (NeurIPS), vol. 32, pp. 15 098\u201315 109 (2019)"}],"container-title":["Lecture Notes in Computer Science","NASA Formal Methods"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-06773-0_16","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T11:11:20Z","timestamp":1659352280000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-06773-0_16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"ISBN":["9783031067723","9783031067730"],"references-count":31,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-06773-0_16","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022]]},"assertion":[{"value":"20 May 2022","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"NFM","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"NASA Formal Methods Symposium","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Pasadena, CA","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"USA","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2022","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"24 May 2022","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"27 May 2022","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"14","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"nfm2022","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/shemesh.larc.nasa.gov\/nfm2022\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Single-blind","order":1,"name":"type","label":"Type","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"EasyChair","order":2,"name":"conference_management_system","label":"Conference Management System","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"118","order":3,"name":"number_of_submissions_sent_for_review","label":"Number of Submissions Sent for Review","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"33","order":4,"name":"number_of_full_papers_accepted","label":"Number of Full Papers Accepted","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"6","order":5,"name":"number_of_short_papers_accepted","label":"Number of Short Papers Accepted","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"28% - The value is computed by the equation \"Number of Full Papers Accepted \/ Number of Submissions Sent for Review * 100\" and then rounded to a whole number.","order":6,"name":"acceptance_rate_of_full_papers","label":"Acceptance Rate of Full Papers","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"3","order":7,"name":"average_number_of_reviews_per_paper","label":"Average Number of Reviews per Paper","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"6.3","order":8,"name":"average_number_of_papers_per_reviewer","label":"Average Number of Papers per Reviewer","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"No","order":9,"name":"external_reviewers_involved","label":"External Reviewers Involved","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}}]}}