{"type":"ScholarlyArticle","title":"Robust Quadratic Programming for Price Optimization","authors":[{"name":"Akihiro Yabe","slug":"akihiro-yabe","given":"Akihiro","family":"Yabe","affiliation":"NEC Corporation"},{"name":"Shinji Ito","slug":"shinji-ito","given":"Shinji","family":"Ito","affiliation":"NEC Corporation"},{"name":"Ryohei Fujimaki","slug":"ryohei-fujimaki","given":"Ryohei","family":"Fujimaki","affiliation":"NEC Corporation"}],"abstract":"The goal of price optimization is to maximize total revenue by adjusting the prices of products, on the basis of predicted sales numbers that are functions of pricing strategies. Recent advances in demand modeling using machine learning raise a new challenge in price \r\noptimization, i.e., how to manage statistical errors in estimation. In this paper, we show that uncertainty in recently-proposed prescriptive price optimization frameworks can be represented by a matrix normal distribution. For this particular uncertainty, we propose novel robust quadratic programming algorithms for conservative lower-bound maximization. We offer an asymptotic probabilistic guarantee of conservativeness of our formulation. Our experiments on both artificial and actual price data show that our robust price optimization allows users to determine best risk-return trade-offs and to explore safe, profitable price strategies.","doi":"10.24963/ijcai.2017/648","url":"https://ijcai.org/proceedings/2017/648","pdf":"https://ijcai.org/proceedings/2017/0648.pdf","markdown":"https://ijcai.org/proceedings/2017/648.md","bibtex":"https://ijcai.org/proceedings/2017/bibtex/648","volume":{"slug":"2017","title":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2017","url":"https://ijcai.org/proceedings/2017/","year":2017,"location":"Melbourne, Australia","dates":"19-25 August 2017","editors":"Carles Sierra","isbn":"978-0-9992411-0-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2017"},"section":"Uncertainty in AI","track":"Main track","number":648,"pages":{"first":4648,"last":4654},"keywords":["Uncertainty in AI: Decision/Utility Theory","Constraints and Satisfiability: Constraint Optimisation"],"videos":[],"references":[{"position":1,"raw":"[Ben-Tal and Nemirovski, 1998] Aharon Ben-Tal and Arkadi Nemirovski. Robust convex optimization. Mathematics of Operations Research, 23(4):769-805, 1998.","title":"Aharon Ben-Tal and Arkadi Nemirovski. Robust convex optimization","authors":"Ben-Tal, Nemirovski","year":"1998","venue":"Mathematics of Operations Research","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Ben-Tal et al., 2002] Aharon Ben-Tal, Arkadi Nemirovski, and Cees Roos. Robust solutions of uncertain quadratic and conic-quadratic problems. SIAM Journal on Optimiza- tion, 13(2):535-560, 2002.","title":"Robust solutions of uncertain quadratic and conic-quadratic problems","authors":"Ben-Tal","year":"2002","venue":"SIAM Journal on Optimization","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Ben-Tal et al., 2009] Aharon Ben-Tal, Laurent El Ghaoui, and Arkadi Nemirovski. Robust optimization. Princeton University Press, 2009.","title":"Robust optimization","authors":"Aharon Ben-Tal, Laurent El Ghaoui, Arkadi Nemirovski","year":"2009","venue":null,"doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Bertsimas et al., 2011] Dimitris Bertsimas, David B. Brown, and Constantine Caramanis. Theory and applica- tions of robust optimization. SIAM review, 53(3):464-501, 2011.","title":"Theory and applications of robust optimization","authors":"Dimitris Bertsimas, David B Brown, Constantine Caramanis","year":"2011","venue":"SIAM review","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Burer and Vandenbussche, 2008] Samuel Burer and Dieter Vandenbussche. A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite re- laxations. Mathematical Programming, 113(2):259-282, 2008.","title":"A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations","authors":"Vandenbussche Burer, Samuel Burer, Dieter Vandenbussche","year":"2008","venue":"Mathematical Programming","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Caro and Gallien, 2012] Felipe Caro and Jérémie Gallien. Clearance pricing optimization for a fast-fashion retailer. Operations Research, 60(6):1404-1422, 2012.","title":"Clearance pricing optimization for a fast-fashion retailer","authors":"Caro, Gallien, Felipe Caro, Jérémie Gallien","year":"2012","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Côté et al., 2003] Jean-Philippe Côté, Patrice Marcotte, and Gilles Savard. A bilevel modelling approach to pricing and fare optimisation in the airline industry. Journal of Revenue and Pricing Management, 2(1):23-36, 2003.","title":"A bilevel modelling approach to pricing and fare optimisation in the airline industry","authors":"Jean-Philippe Côté, Patrice Marcotte, Gilles Savard","year":"2003","venue":"Journal of Revenue and Pricing Management","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Delage and Ye, 2010] Erick Delage and Yinyu Ye. Distri- butionally robust optimization under moment uncertainty with application to data-driven problems. Operations re- search, 58(3):595-612, 2010.","title":"Erick Delage and Yinyu Ye. Distributionally robust optimization under moment uncertainty with application to data-driven problems","authors":"Ye Delage","year":"2010","venue":"Operations research","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Duchi et al., 2016] John C. Duchi, Peter W. Glynn, and Hongseok Namkoong. Statistics of robust optimization: A generalized empirical likelihood approach. Stanford Uni- versity, 2016.","title":"Statistics of robust optimization: A generalized empirical likelihood approach","authors":"John C Duchi, Peter W Glynn, Hongseok Namkoong","year":"2016","venue":null,"doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Ferreira et al., 2015] Kris J. Ferreira, Bin H. A. Lee, and David Simchi-Levi. Analytics for an online retailer: De- mand forecasting and price optimization. Manufacturing & Service Operations Management, 2015.","title":"Analytics for an online retailer: Demand forecasting and price optimization","authors":"Kris J Ferreira, H A Bin, David Lee, Simchi-Levi","year":"2015","venue":"Manufacturing & Service Operations Management","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Ghaoui and Lebret, 1997] Laurent El Ghaoui and Hervé Le- bret. Robust solutions to least-squares problems with un- certain data. SIAM Journal on Matrix Analysis and Appli- cations, 18(4):1035-1064, 1997.","title":"Laurent El Ghaoui and Hervé Lebret. Robust solutions to least-squares problems with uncertain data","authors":"Lebret Ghaoui","year":"1997","venue":"SIAM Journal on Matrix Analysis and Applications","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Gupta and Nagar, 1999] Arjun K. Gupta and Daya K. Na- gar. Matrix variate distributions, volume 104. CRC Press, 1999.","title":"Matrix variate distributions","authors":"K Arjun, Gupta, K Daya, Nagar","year":"1999","venue":null,"doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Halldórsson and Tütüncü, 2003] Bjarni V. Halldórsson and Reha H. Tütüncü. An interior-point method for a class of saddle-point problems. Journal of Optimization Theory and Applications, 116(3):559-590, 2003.","title":"An interior-point method for a class of saddle-point problems","authors":"V Bjarni, Reha H Halldórsson, Tütüncü","year":"2003","venue":"Journal of Optimization Theory and Applications","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Ito and Fujimaki, 2016] Shinji Ito and Ryohei Fujimaki. Large-scale price optimization via network flow. In Ad- vances in Neural Information Processing Systems, 2016.","title":"Large-scale price optimization via network flow","authors":"Fujimaki ; ] Shinji Ito, Ryohei Ito, Fujimaki","year":"2016","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Ito and Fujimaki, 2017] Shinji Ito and Ryohei Fujimaki. Optimization beyond prediction: Prescriptive price opti- mization. In Proceedings of the 23nd ACM SIGKDD In- ternational Conference on Knowledge Discovery and Data Mining. ACM, 2017.","title":"Optimization beyond prediction: Prescriptive price optimization","authors":"Fujimaki ; ] Shinji Ito, Ryohei Ito, Fujimaki","year":"2017","venue":"Proceedings of the 23nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Kiefer, 1953] Jack Kiefer. Sequential minimax search for a maximum. Proceedings of the American Mathematical Society, 4(3):502-506, 1953.","title":"Sequential minimax search for a maximum","authors":"Jack Kiefer, Kiefer","year":"1953","venue":"Proceedings of the American Mathematical Society","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Kolmogorov and Zabin, 2004] Vladimir Kolmogorov and Ramin Zabin. What energy functions can be minimized via graph cuts? Pattern Analysis and Machine Intelli- gence, IEEE Transactions on, 26(2):147-159, 2004.","title":"What energy functions can be minimized via graph cuts? Pattern Analysis and Machine Intelligence","authors":"Zabin Kolmogorov","year":"2004","venue":"IEEE Transactions on","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Koushik et al., 2012] Dev Koushik, Jon A. Higbie, and Craig Eister. Retail price optimization at intercontinental hotels group. Interfaces, 42(1):45-57, 2012.","title":"Retail price optimization at intercontinental hotels group","authors":"Koushik","year":"2012","venue":"Interfaces","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Kunz and Crone, 2014] Timo P. Kunz and Sven F. Crone. Demand models for the static retail price optimization problem-a revenue management perspective. In OASIcs- OpenAccess Series in Informatics, volume 37. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014.","title":"Demand models for the static retail price optimization problem-a revenue management perspective","authors":"P Timo, Sven F Kunz, Crone","year":"2014","venue":"OASIcs-OpenAccess Series in Informatics","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Lee, 2011] Seonah Lee. Study of demand models and price optimization performance. PhD thesis, Georgia Institute of Technology, 2011.","title":"Seonah Lee. Study of demand models and price optimization performance","authors":"Lee","year":"2011","venue":null,"doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Marshall, 2009] Alfred Marshall. Principles of economics: unabridged eighth edition. Cosimo, Inc., 2009.","title":"Principles of economics: unabridged eighth edition","authors":"Marshall","year":"2009","venue":null,"doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Nemirovski, 1999] Arkadi Nemirovski. On self-concordant convex-concave functions. Optimization Methods and Software, 11(1-4):303-384, 1999.","title":"On self-concordant convex-concave functions","authors":"Arkadi Nemirovski","year":"1999","venue":"Optimization Methods and Software","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Tütüncü and Koenig, 2004] R. H. Tütüncü and M. Koenig. Robust asset allocation. Annals of Operations Research, 132(1-4):157-187, 2004.","title":"Robust asset allocation","authors":"R H Tütüncü, M Koenig","year":"2004","venue":"Annals of Operations Research","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[van Ryzin and Mahajan, 1999] Garrett van Ryzin and Sid- dharth Mahajan. On the relationship between inventory costs and variety benefits in retail assortments. Manage- ment Science, 45(11):1496-1509, 1999.","title":"On the relationship between inventory costs and variety benefits in retail assortments","authors":"Siddharth Van Ryzin And Mahajan ; Ryzin, Mahajan","year":"1999","venue":"Management Science","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Wang et al., 2015] Jialei Wang, Ryohei Fujimaki, and Yosuke Motohashi. Trading interpretability for accu- racy: Oblique treed sparse additive models. In Proceed- ings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1245- 1254. ACM, 2015.","title":"Trading interpretability for accuracy: Oblique treed sparse additive models","authors":"Wang","year":"2015","venue":"Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}