{"type":"ScholarlyArticle","title":"Group Sparse Optimal Transport for Sparse Process Flexibility Design","authors":[{"name":"Dixin Luo","slug":"dixin-luo","given":"Dixin","family":"Luo","affiliation":"School of Computer Science and Technology , Beijing Institue of Technology"},{"name":"Tingting Yu","slug":"tingting-yu","given":"Tingting","family":"Yu","affiliation":"School of Computer Science and Technology , Beijing Institue of Technology"},{"name":"Hongteng Xu","slug":"hongteng-xu","given":"Hongteng","family":"Xu","affiliation":"Gaoling School of Artificial Intelligence , Renmin University of China; Beijing Key Laboratory of Big Data Management and Analysis Methods"}],"abstract":"As a fundamental problem in Operations Research, sparse process flexibility design (SPFD) aims to design a manufacturing network across industries that achieves a trade-off between the efficiency and robustness of supply chains. \r\nIn this study, we propose a novel solution to this problem with the help of computational optimal transport techniques.\r\nGiven a set of supply-demand pairs, we formulate the SPFD task approximately as a group sparse optimal transport (GSOT) problem, in which a group of couplings between the supplies and demands is optimized with a group sparse regularizer. \r\nWe solve this optimization problem via an algorithmic framework of alternating direction method of multipliers (ADMM), in which the target network topology is updated by soft-thresholding shrinkage, and the couplings of the OT problems are updated via a smooth OT algorithm in parallel. \r\nThis optimization algorithm has guaranteed convergence and provides a generalized framework for the SPFD task, which is applicable regardless of whether the supplies and demands are balanced. \r\nExperiments show that our GSOT-based method can outperform representative heuristic methods in various SPFD tasks.\r\nAdditionally, when implementing the GSOT method, the proposed ADMM-based optimization algorithm is comparable or superior to the commercial software Gurobi. \r\nThe code is available at https://github.com/Dixin-s-Lab/GSOT.","doi":"10.24963/ijcai.2023/679","url":"https://ijcai.org/proceedings/2023/679","pdf":"https://ijcai.org/proceedings/2023/0679.pdf","markdown":"https://ijcai.org/proceedings/2023/679.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/679","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"AI for Good","track":"AI for Good","number":679,"pages":{"first":6121,"last":6129},"keywords":["AI for Good: Multidisciplinary Topics and Applications"],"videos":[],"references":[{"position":1,"raw":"[Agueh and Carlier, 2011] Martial Agueh and Guillaume Carlier. Barycenters in the Wasserstein space. SIAM Jour- nal on Mathematical Analysis, 43(2):904-924, 2011.","title":"Martial Agueh and Guillaume Carlier. Barycenters in the Wasserstein space","authors":"Carlier Agueh","year":"2011","venue":"SIAM Journal on Mathematical Analysis","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Altschuler et al., 2017] Jason Altschuler, Jonathan Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. In Advances in Neural Information Processing Systems, pages 1964-1974, 2017.","title":"Near-linear time approximation algorithms for optimal transport via sinkhorn iteration","authors":"Jason Altschuler, Jonathan Weed, Philippe Rigollet","year":"2017","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Arjovsky et al., 2017] Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial net- works. In International Conference on Machine Learning, pages 214-223, 2017.","title":"Wasserstein generative adversarial networks","authors":"Martin Arjovsky, Soumith Chintala, Léon Bottou","year":"2017","venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Barrett et al., 2020] Thomas Barrett, William Clements, Jakob Foerster, and Alex Lvovsky. Exploratory combi- natorial optimization with reinforcement learning. In Pro- ceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 3243-3250, 2020.","title":"Exploratory combinatorial optimization with reinforcement learning","authors":"Thomas Barrett, William Clements, Jakob Foerster, Alex Lvovsky","year":"2020","venue":"Proceedings of the AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Beck and Teboulle, 2009] Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for lin- ear inverse problems. SIAM journal on imaging sciences, 2(1):183-202, 2009.","title":"A fast iterative shrinkage-thresholding algorithm for linear inverse problems","authors":"Amir Beck, Marc Teboulle","year":"2009","venue":"SIAM journal on imaging sciences","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Benamou et al., 2015] Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, and Gabriel Peyré. Iterative Bregman projections for regularized transporta- tion problems. SIAM Journal on Scientific Computing, 37(2):A1111-A1138, 2015.","title":"Iterative Bregman projections for regularized transportation problems","authors":"Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, Gabriel Peyré","year":"2015","venue":"SIAM Journal on Scientific Computing","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Bengio et al., 2009] Samy Bengio, Fernando Pereira, Yoram Singer, and Dennis Strelow. Group sparse coding. Advances in neural information processing systems, 22, 2009.","title":"Group sparse coding","authors":"Samy Bengio, Fernando Pereira, Yoram Singer, Dennis Strelow","year":"2009","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Blondel et al., 2018] Mathieu Blondel, Vivien Seguy, and Antoine Rolet. Smooth and sparse optimal transport. In In- ternational conference on artificial intelligence and statis- tics, pages 880-889. PMLR, 2018.","title":"Smooth and sparse optimal transport","authors":"Mathieu Blondel, Vivien Seguy, Antoine Rolet","year":"2018","venue":"International conference on artificial intelligence and statistics","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Boyd et al., 2011] Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. Distributed op- timization and statistical learning via the alternating direc- tion method of multipliers. Foundations and Trends® in Machine learning, 3(1):1-122, 2011.","title":"Distributed optimization and statistical learning via the alternating direction method of multipliers","authors":"Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein","year":"2011","venue":"Foundations and Trends® in Machine learning","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Chan et al., 2022] Timothy CY Chan, Daniel Letourneau, and Benjamin G Potter. Sparse flexible design: a machine learning approach. Flexible Services and Manufacturing Journal, pages 1-51, 2022.","title":"Sparse flexible design: a machine learning approach","authors":"C Y Timothy, Daniel Chan, Benjamin G Letourneau, Potter","year":"2022","venue":"Flexible Services and Manufacturing Journal","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Chen et al., 2015] Xi Chen, Jiawei Zhang, and Yuan Zhou. Optimal sparse designs for process flexibility via prob- abilistic expanders. Operations Research, 63(5):1159- 1176, 2015.","title":"Optimal sparse designs for process flexibility via probabilistic expanders","authors":"Xi Chen, Jiawei Zhang, Yuan Zhou","year":"2015","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Chen et al., 2019] Xi Chen, Tengyu Ma, Jiawei Zhang, and Yuan Zhou. Optimal design of process flexibility for gen- eral production systems. Operations Research, 67(2):516- 531, 2019.","title":"Optimal design of process flexibility for general production systems","authors":"Xi Chen, Tengyu Ma, Jiawei Zhang, Yuan Zhou","year":"2019","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Chizat et al., 2018] Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, and Franc ¸ois-Xavier Vialard. Scaling algo- rithms for unbalanced optimal transport problems. Math- ematics of Computation, 87(314):2563-2609, 2018.","title":"Scaling algo-rithms for unbalanced optimal transport problems","authors":"Chizat","year":"2018","venue":"Mathematics of Computation","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Chou et al., 2008] Mabel C Chou, Chung-Piaw Teo, and Huan Zheng. Process flexibility: design, evaluation, and applications. Flexible Services and Manufacturing Jour- nal, 20:59-94, 2008.","title":"Process flexibility: design, evaluation, and applications","authors":"C Mabel, Chung-Piaw Chou, Huan Teo, Zheng","year":"2008","venue":"Flexible Services and Manufacturing Journal","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Chou et al., 2010] Mabel C Chou, Geoffrey A Chua, Chung-Piaw Teo, and Huan Zheng. Design for process flexibility: Efficiency of the long chain and sparse struc- ture. Operations research, 58(1):43-58, 2010.","title":"Design for process flexibility: Efficiency of the long chain and sparse structure","authors":"C Mabel, Geoffrey A Chou, Chung-Piaw Chua, Huan Teo, Zheng","year":"2010","venue":"Operations research","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Chou et al., 2011] Mabel C Chou, Geoffrey A Chua, Chung-Piaw Teo, and Huan Zheng. Process flexibility re- visited: The graph expander and its applications. Opera- tions research, 59(5):1090-1105, 2011.","title":"Process flexibility revisited: The graph expander and its applications","authors":"C Mabel, Geoffrey A Chou, Chung-Piaw Chua, Huan Teo, Zheng","year":"2011","venue":"Operations research","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Chou et al., 2014] Mabel C Chou, Geoffrey A Chua, and Huan Zheng. On the performance of sparse process struc- tures in partial postponement production systems. Opera- tions research, 62(2):348-365, 2014.","title":"On the performance of sparse process structures in partial postponement production systems","authors":"C Mabel, Geoffrey A Chou, Huan Chua, Zheng","year":"2014","venue":"Operations research","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Cuturi and Doucet, 2014] Marco Cuturi and Arnaud Doucet. Fast computation of Wasserstein barycenters. In International Conference on Machine Learning, pages 685-693, 2014.","title":"Fast computation of Wasserstein barycenters","authors":"Marco Cuturi, Arnaud Doucet","year":"2014","venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Cuturi, 2013] Marco Cuturi. Sinkhorn distances: Light- speed computation of optimal transport. In Advances in neural information processing systems, pages 2292-2300, 2013.","title":"Sinkhorn distances: Lightspeed computation of optimal transport","authors":"Marco Cuturi","year":"2013","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Deng et al., 2013] Wei Deng, Wotao Yin, and Yin Zhang. Group sparse optimization by alternating direction method. In Wavelets and Sparsity XV, volume 8858, pages 242-256. SPIE, 2013.","title":"Group sparse optimization by alternating direction method","authors":"Wei Deng, Wotao Yin, Yin Zhang","year":"2013","venue":"Wavelets and Sparsity XV","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Désir et al., 2016] Antoine Désir, Vineet Goyal, Yehua Wei, and Jiawei Zhang. Sparse process flexibility designs: Is the long chain really optimal? Operations Research, 64(2):416-431, 2016.","title":"Sparse process flexibility designs: Is the long chain really optimal?","authors":"Antoine Désir, Vineet Goyal, Yehua Wei, Jiawei Zhang","year":"2016","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Feng et al., 2017] Wancheng Feng, Chen Wang, and Zuo- Jun Max Shen. Process flexibility design in heterogeneous and unbalanced networks: A stochastic programming ap- proach. IISE Transactions, 49(8):781-799, 2017.","title":"Process flexibility design in heterogeneous and unbalanced networks: A stochastic programming approach","authors":"Wancheng Feng, Chen Wang, Zuo-Jun Max Shen","year":"2017","venue":"IISE Transactions","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Flamary et al., 2021] Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z. Alaya, Aurélie Bois- bunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, Léo Gautheron, Nathalie T.H. Gayraud, Hicham Janati, Alain Rakotoma- monjy, Ievgen Redko, Antoine Rolet, Antony Schutz, Vivien Seguy, Danica J. Sutherland, Romain Tavenard, Alexander Tong, and Titouan Vayer. Pot: Python opti- mal transport. Journal of Machine Learning Research, 22(78):1-8, 2021.","title":"Pot: Python optimal transport","authors":"Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Z Mokhtar, Aurélie Alaya, Stanislas Boisbunon, Laetitia Chambon, Adrien Chapel, Kilian Corenflos, Nemo Fatras, Léo Fournier, Nathalie T H Gautheron, Hicham Gayraud, Alain Janati, Ievgen Rakotomamonjy, Antoine Redko, Antony Rolet, Vivien Schutz, Danica J Seguy, Romain Sutherland, Alexander Tavenard, Titouan Tong, Vayer","year":"2021","venue":"Journal of Machine Learning Research","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Frogner et al., 2015] Charlie Frogner, Chiyuan Zhang, Hos- sein Mobahi, Mauricio Araya, and Tomaso A Poggio. Learning with a wasserstein loss. In Advances in Neural Information Processing Systems, pages 2053-2061, 2015.","title":"Learning with a wasserstein loss","authors":"Charlie Frogner, Chiyuan Zhang, Hossein Mobahi, Mauricio Araya, Tomaso A Poggio","year":"2015","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Graves and Tomlin, 2003] Stephen C Graves and Brian T Tomlin. Process flexibility in supply chains. Management Science, 49(7):907-919, 2003.","title":"Process flexibility in supply chains","authors":"C Stephen, Brian T Graves, Tomlin","year":"2003","venue":"Management Science","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Gurobi Optimization, 2021] LLC Gurobi Optimization. Gurobi optimizer reference manual, 2021.","title":"LLC Gurobi Optimization. Gurobi optimizer reference manual","authors":null,"year":"2021","venue":"Gurobi Optimization","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Harris and Ross, 1956] TE Harris and FS Ross. Fundamen- tals of a method for evaluating rail net capacities. Research Memorandum, 1573, 1956.","title":"Fundamentals of a method for evaluating rail net capacities","authors":"Ross ; Te Harris, F S Harris, Ross","year":"1956","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Hughes and Chen, 2021] Jason Hughes and Juntao Chen. Fair and distributed dynamic optimal transport for re- source allocation over networks. In 2021 55th Annual Conference on Information Sciences and Systems (CISS), pages 1-6. IEEE, 2021.","title":"Fair and distributed dynamic optimal transport for resource allocation over networks","authors":"Chen ; Hughes, Jason Hughes, Juntao Chen","year":"2021","venue":"55th Annual Conference on Information Sciences and Systems (CISS)","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Jordan and Graves, 1995] William C Jordan and Stephen C Graves. Principles on the benefits of manufacturing pro- cess flexibility. Management science, 41(4):577-594, 1995.","title":"Principles on the benefits of manufacturing process flexibility","authors":"C William, Stephen C Jordan, Graves","year":"1995","venue":"Management science","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Khalil et al., 2017] Elias Khalil, Hanjun Dai, Yuyu Zhang, Bistra Dilkina, and Le Song. Learning combinatorial op- timization algorithms over graphs. Advances in neural in- formation processing systems, 30, 2017.","title":"Learning combinatorial optimization algorithms over graphs","authors":"Elias Khalil, Hanjun Dai, Yuyu Zhang, Bistra Dilkina, Le Song","year":"2017","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Kusner et al., 2015] Matt Kusner, Yu Sun, Nicholas Kolkin, and Kilian Weinberger. From word embeddings to docu- ment distances. In International conference on machine learning, pages 957-966. PMLR, 2015.","title":"From word embeddings to document distances","authors":"Matt Kusner, Yu Sun, Nicholas Kolkin, Kilian Weinberger","year":"2015","venue":"International conference on machine learning","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Lien et al., 2011] Robert W Lien, Seyed MR Iravani, Karen Smilowitz, and Michal Tzur. An efficient and robust de- sign for transshipment networks. Production and Opera- tions Management, 20(5):699-713, 2011.","title":"An efficient and robust design for transshipment networks","authors":"Robert W Lien, M R Seyed, Karen Iravani, Michal Smilowitz, Tzur","year":"2011","venue":"Production and Operations Management","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Liu and Nocedal, 1989] Dong C Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimiza- tion. Mathematical programming, 45(1-3):503-528, 1989.","title":"On the limited memory bfgs method for large scale optimization","authors":"C Dong, Jorge Liu, Nocedal","year":"1989","venue":"Mathematical programming","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Maretic et al., 2022] Hermina Petric Maretic, Mireille El Gheche, Matthias Minder, Giovanni Chierchia, and Pascal Frossard. Wasserstein-based graph alignment. IEEE Transactions on Signal and Information Processing over Networks, 8:353-363, 2022.","title":"Wasserstein-based graph alignment","authors":"Hermina Petric Maretic, Mireille El Gheche, Matthias Minder, Giovanni Chierchia, Pascal Frossard","year":"2022","venue":"IEEE Transactions on Signal and Information Processing over Networks","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Nishihara et al., 2015] Robert Nishihara, Laurent Lessard, Ben Recht, Andrew Packard, and Michael Jordan. A gen- eral analysis of the convergence of admm. In International conference on machine learning, pages 343-352. PMLR, 2015.","title":"A general analysis of the convergence of admm","authors":"Robert Nishihara, Laurent Lessard, Ben Recht, Andrew Packard, Michael Jordan","year":"2015","venue":"International conference on machine learning","doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Peyré et al., 2019] Gabriel Peyré, Marco Cuturi, et al. Com- putational optimal transport. Foundations and Trends® in Machine Learning, 11(5-6):355-607, 2019.","title":"Computational optimal transport","authors":"Gabriel Peyré, Marco Cuturi","year":"2019","venue":"Foundations and Trends® in Machine Learning","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Schölkopf et al., 2001] Bernhard Schölkopf, Ralf Herbrich, and Alex J Smola. A generalized representer theorem. In Proceedings of the 14th Annual Conference on Computa- tional Learning Theory and and 5th European Conference on Computational Learning Theory, pages 416-426, 2001.","title":"A generalized representer theorem","authors":"Bernhard Schölkopf, Ralf Herbrich, Alex J Smola","year":"2001","venue":"Proceedings of the 14th Annual Conference on Computational Learning Theory and and 5th European Conference on Computational Learning Theory","doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Simchi-Levi and Wei, 2012] David Simchi-Levi and Yehua Wei. Understanding the performance of the long chain and sparse designs in process flexibility. Operations research, 60(5):1125-1141, 2012.","title":"Understanding the performance of the long chain and sparse designs in process flexibility","authors":"Simchi-Levi And Wei, ; ] David Simchi-Levi, Yehua Wei","year":"2012","venue":"Operations research","doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Simchi-Levi et al., 2018] David Simchi-Levi, He Wang, and Yehua Wei. Increasing supply chain robustness through process flexibility and inventory. Production and Operations Management, 27(8):1476-1491, 2018.","title":"Increasing supply chain robustness through process flexibility and inventory","authors":"David Simchi-Levi, He Wang, Yehua Wei","year":"2018","venue":"Production and Operations Management","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Simon et al., 2013] Noah Simon, Jerome Friedman, Trevor Hastie, and Robert Tibshirani. A sparse-group lasso. Jour- nal of computational and graphical statistics, 22(2):231- 245, 2013.","title":"A sparse-group lasso","authors":"Simon","year":"2013","venue":"Journal of computational and graphical statistics","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"[Sinkhorn and Knopp, 1967] Richard Sinkhorn and Paul Knopp. Concerning nonnegative matrices and doubly stochastic matrices. Pacific Journal of Mathematics, 21(2):343-348, 1967.","title":"Richard Sinkhorn and Paul Knopp. Concerning nonnegative matrices and doubly stochastic matrices","authors":"Knopp Sinkhorn","year":"1967","venue":"Pacific Journal of Mathematics","doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Sultana et al., 2022] Nasrin Sultana, Jeffrey Chan, Tabinda Sarwar, and AK Qin. Learning to optimise general tsp instances. International Journal of Machine Learning and Cybernetics, 13(8):2213-2228, 2022.","title":"Learning to optimise general tsp instances","authors":"Nasrin Sultana, Jeffrey Chan, Tabinda Sarwar, Qin","year":"2022","venue":"International Journal of Machine Learning and Cybernetics","doi":null,"url":null,"internal_url":null},{"position":43,"raw":"[Titouan et al., 2019] Vayer Titouan, Nicolas Courty, Ro- main Tavenard, and Rémi Flamary. Optimal transport for structured data with application on graphs. In Inter- national Conference on Machine Learning, pages 6275- 6284. PMLR, 2019.","title":"Optimal transport for structured data with application on graphs","authors":"Nicolas Vayer Titouan, Romain Courty, Rémi Tavenard, Flamary","year":"2019","venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":44,"raw":"[Villani, 2008] Cédric Villani. Optimal transport: Old and new, volume 338. Springer Science & Business Media, 2008.","title":"Optimal transport: Old and new","authors":"Villani","year":"2008","venue":null,"doi":null,"url":null,"internal_url":null},{"position":45,"raw":"[Wang and Banerjee, 2014] Huahua Wang and Arindam Banerjee. Bregman alternating direction method of mul- tipliers. In Advances in Neural Information Processing Systems, pages 2816-2824, 2014.","title":"Bregman alternating direction method of multipliers","authors":"Wang, Banerjee, Huahua Wang, Arindam Banerjee","year":"2014","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":46,"raw":"[Wei et al., 2021] Yehua Wei, Lei Zhang, Ruiyi Zhang, Shi- jing Si, Hao Zhang, and Lawrence Carin. Reinforcement learning for flexibility design problems. arXiv preprint arXiv:2101.00355, 2021.","title":"Reinforcement learning for flexibility design problems","authors":"Yehua Wei, Lei Zhang, Ruiyi Zhang, Shijing Si, Hao Zhang, Lawrence Carin","year":"2021","venue":null,"doi":null,"url":null,"internal_url":null},{"position":47,"raw":"[Xie et al., 2020] Yujia Xie, Xiangfeng Wang, Ruijia Wang, and Hongyuan Zha. A fast proximal point method for com- puting exact wasserstein distance. In Uncertainty in artifi- cial intelligence, pages 433-453. PMLR, 2020.","title":"A fast proximal point method for computing exact wasserstein distance","authors":"Yujia Xie, Xiangfeng Wang, Ruijia Wang, Hongyuan Zha","year":"2020","venue":"Uncertainty in artificial intelligence","doi":null,"url":null,"internal_url":null},{"position":48,"raw":"[Xu and Cheng, 2022] Hongteng Xu and Minjie Cheng. Regularized optimal transport layers for gener- alized global pooling operations. arXiv preprint arXiv:2212.06339, 2022.","title":"Regularized optimal transport layers for generalized global pooling operations","authors":"Cheng Xu, Hongteng Xu, Minjie Cheng","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":49,"raw":"[Yan et al., 2018] Zhenzhen Yan, Sarah Yini Gao, and Chung Piaw Teo. On the design of sparse but ef- ficient structures in operations. Management Science, 64(7):3421-3445, 2018.","title":"On the design of sparse but efficient structures in operations","authors":"Zhenzhen Yan, Sarah Yini Gao, Chung Piaw Teo","year":"2018","venue":"Management Science","doi":null,"url":null,"internal_url":null},{"position":50,"raw":"[Ye et al., 2017] Jianbo Ye, Panruo Wu, James Z Wang, and Jia Li. Fast discrete distribution clustering using wasser- stein barycenter with sparse support. IEEE Transactions on Signal Processing, 65(9):2317-2332, 2017.","title":"Fast discrete distribution clustering using wasserstein barycenter with sparse support","authors":"Jianbo Ye, Panruo Wu, James Z Wang, Jia Li","year":"2017","venue":"IEEE Transactions on Signal Processing","doi":null,"url":null,"internal_url":null},{"position":51,"raw":"[Zhang and Zhu, 2019] Rui Zhang and Quanyan Zhu. Consensus-based distributed discrete optimal transport for decentralized resource matching. IEEE Transactions on Signal and Information Processing over Networks, 5(3):511-524, 2019.","title":"Consensus-based distributed discrete optimal transport for decentralized resource matching","authors":"Rui Zhang, Quanyan Zhu","year":"2019","venue":"IEEE Transactions on Signal and Information Processing over Networks","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}