{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T09:52:46Z","timestamp":1788861166329,"version":"build-2803163510"},"reference-count":45,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100006234","name":"Sandia National Laboratories","doi-asserted-by":"publisher","award":["DE-NA-0003525"],"award-info":[{"award-number":["DE-NA-0003525"]}],"id":[{"id":"10.13039\/100006234","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>Distributionally robust optimization (DRO) is widely used because it offers a way to overcome the conservativeness of robust optimization without requiring the specificity of stochastic programming. On the computational side, many practical DRO instances can be equivalently (or approximately) formulated as semidefinite programming (SDP) problems via conic duality of the moment problem. However, despite being theoretically solvable in polynomial time, SDP problems in practice are computationally challenging and quickly become intractable with increasing problem sizes. We propose a new approximation method to solve DRO problems with moment-based ambiguity sets. Our approximation method relies on principal component analysis (PCA) for optimal lower dimensional representation of variability in random samples. We show that the PCA approximation yields a relaxation of the original problem and derive theoretical bounds on the gap between the original problem and its PCA approximation. Furthermore, an extensive numerical study shows the strength of the proposed approximation method in terms of solution quality and runtime. As examples, for distributionally robust conditional value-at-risk and risk-averse production-transportation problems the proposed PCA approximation using only 50% of the principal components yields near-optimal solutions (within 1%) with a one to two order of magnitude reduction in computation time.<\/jats:p>","DOI":"10.1137\/16m1075910","type":"journal-article","created":{"date-parts":[[2018,6,14]],"date-time":"2018-06-14T12:29:58Z","timestamp":1528979398000},"page":"1817-1841","source":"Crossref","is-referenced-by-count":12,"title":["Distributionally Robust Optimization with Principal Component Analysis"],"prefix":"10.1137","volume":"28","author":[{"given":"Jianqiang","family":"Cheng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard","family":"Li-Yang Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Habib N.","family":"Najm","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali","family":"Pinar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cosmin","family":"Safta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Paul","family":"Watson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,14]]},"reference":[{"key":"atypb1","first-page":"1","author":"Bayraksan G.","year":"2015","journal-title":"INFORMS"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"A. Ben-Tal, L. El Ghaoui, and A. Nemirovski,\n                      Robust Optimization\n                      , Princeton University Press, Princeton, NJ, 2009.","DOI":"10.1515\/9781400831050"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/080734510"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TPWRS.2012.2205021"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/130910312"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1100.0445"},{"key":"atypb7","unstructured":"J. Blanchet, Y. Kang, and K. Murthy,\n                      Robust Wasserstein Profile Inference and Applications to Machine Learning\n                      , preprint, arXiv:1610.05627, 2016."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/130915315"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2014.05.009"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1090.0741"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02288367"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-005-0678-0"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"P. M. Esfahani and D. Kuhn,\n                      Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations\n                      , Math. Program. (2017),https:\/\/doi.org\/10.1007\/s10107-017-1172-1.","DOI":"10.1007\/s10107-017-1172-1"},{"key":"atypb14","unstructured":"R. Gao, X. Chen, and A. J. Kleywegt,\n                      Distributional Robustness and Regularization in Statistical Learning\n                      , preprint, arXiv:1712.06050, 2017."},{"key":"atypb15","unstructured":"R. Gao and A. J. Kleywegt,\n                      Distributionally Robust Stochastic Optimization with Wasserstein Distance\n                      , preprint, arXiv:1604.02199, 2016."},{"key":"atypb16","unstructured":"R. Gao and A. J. Kleywegt,\n                      Distributionally Robust Stochastic Optimization with Dependence Structure\n                      , preprint, arXiv:1701.04200, 2017."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"J. Gotoh, M. J. Kim, and A. Lim,\n                      Robust Empirical Optimization Is Almost the Same as Mean-Variance Optimization\n                      , preprint,https:\/\/ssrn.com\/abstract=3073013, 2015.","DOI":"10.2139\/ssrn.2827400"},{"key":"atypb18","unstructured":"M. Grant and S. Boyd,\n                      CVX: Matlab Software for Disciplined Convex Programming\n                      , Version 2.0 Beta,http:\/\/cvxr.com\/cvx, September 2013."},{"key":"atypb19","first-page":"95","author":"Grant M.","year":"2008","journal-title":"London"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0896-z"},{"key":"atypb21","unstructured":"High Dividend Stocks by Sector\n                      ,http:\/\/www.doubledividendstocks.com, August 26, 2015."},{"key":"atypb22","unstructured":"R. A. Horn and C. R. Johnson,\n                      Matrix Analysis\n                      , Cambridge University Press, Cambridge, 2012."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02868641"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0929-7"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1109\/9.50357"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2015.0776"},{"key":"atypb27","unstructured":"B. Lindgren,\n                      Statistical Theory\n                      , 4th ed., Chapman & Hall\/CRC Texts Statis. Sci. 22, CRC Press, New York, 1993."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-016-1019-1"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.2010.00417.x"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/050622328"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1040.0137"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1060.0353"},{"key":"atypb33","unstructured":"A. Pre\u0301kopa,\n                      Stochastic Programming\n                      , Kluwer Academic Publishers, Dordrecht, 1995."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.21314\/JOR.2000.038"},{"key":"atypb35","unstructured":"H. Scarf,\n                      A min-max solution of an inventory problem\n                      , in Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Redwood City, CA, 1958, pp. 201-209."},{"key":"atypb36","first-page":"1576","author":"Shafieezadeh-Abadeh S.","year":"2015","journal-title":"NY"},{"key":"atypb37","unstructured":"S. Shafieezadeh-Abadeh, D. Kuhn, and P. M. Esfahani,\n                      Regularization via Mass Transportation\n                      , preprint, arXiv:1710.10016, 2017."},{"key":"atypb38","first-page":"165","volume":"135","author":"Shapiro A.","year":"2001","journal-title":"New York"},{"key":"atypb39","doi-asserted-by":"crossref","unstructured":"A. Shapiro, D. Dentcheva, and A. Ruszczyn\u0301ski,\n                      Lectures on stochastic programming: Modeling and theory\n                      , MOS-SIAM Ser. Optim. 16, SIAM, Philadelphia, PA, 2014.","DOI":"10.1137\/1.9781611973433"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0878-1"},{"key":"atypb41","doi-asserted-by":"crossref","unstructured":"B. P. G. Van Parys, P. J. Goulart, and M. Morari,\n                      Distributionally robust expectation inequalities for structured distributions\n                      , Math. Program. (2017),https:\/\/doi.org\/10.1007\/s10107-017-1220-x.","DOI":"10.1007\/s10107-017-1220-x"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/0169-7439(87)80084-9"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2014.1314"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0494-7"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-002-0337-7"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1075910","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:04:01Z","timestamp":1787339041000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1075910"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M1075910"],"URL":"https:\/\/doi.org\/10.1137\/16m1075910","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}