{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:35:03Z","timestamp":1787330103440,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1613170"],"award-info":[{"award-number":["DMS-1613170"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We introduce high order Bellman equations, extending classical Bellman equations to the tensor setting. We introduce weakly chained diagonally dominant (w.c.d.d.) tensors and show that a sufficient condition for the existence and uniqueness of a positive solution to a high order Bellman equation is that the tensors appearing in the equation are w.c.d.d. M-tensors. In this case, we give a policy iteration algorithm to compute this solution. We also prove that a weakly diagonally dominant Z-tensor with nonnegative diagonals is a strong M-tensor if and only if it is w.c.d.d. This last point is analogous to a corresponding result in the matrix setting and tightens a result from [L. Zhang, L. Qi, and G. Zhou, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 437--452]. We apply our results to obtain a provably convergent numerical scheme for an optimal control problem using an \u201coptimize then discretize\u201d approach which outperforms (in both computation time and accuracy) a classical \u201cdiscretize then optimize\u201d approach. To the best of our knowledge, a link between M-tensors and optimal control has not been previously established.<\/jats:p>","DOI":"10.1137\/18m1196923","type":"journal-article","created":{"date-parts":[[2019,2,19]],"date-time":"2019-02-19T10:15:58Z","timestamp":1550571358000},"page":"276-298","source":"Crossref","is-referenced-by-count":22,"title":["High Order Bellman Equations and Weakly Chained Diagonally Dominant Tensors"],"prefix":"10.1137","volume":"40","author":[{"given":"Parsiad","family":"Azimzadeh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1926-4570","authenticated-orcid":true,"given":"Erhan","family":"Bayraktar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,2,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3347"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/15M1043431"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-1991-4305"},{"key":"atypb4","first-page":"140","author":"Betts J. T.","year":"2005","journal-title":"Philadelphia"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/08073041X"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1964431117"},{"key":"atypb7","first-page":"176","author":"Chen Y.","year":"2018","journal-title":"J. Sci. Comput."},{"key":"atypb8","first-page":"1","volume":"27","author":"Crandall M. G.","year":"1992","journal-title":"S.)"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.08.038"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-015-0156-7"},{"key":"atypb11","first-page":"11","author":"Forsyth P. A.","year":"2007","journal-title":"J. Comput. Finance"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.02.042"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-013-4752-4"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"H. J. Kushner and P. G. Dupuis,\n                      Numerical methods for stochastic control problems in continuous time\n                      , Appl. Math. 24, Springer, New York, 1992,https:\/\/doi.org\/10.1007\/978-1-4684-0441-8.","DOI":"10.1007\/978-1-4684-0441-8"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903435235"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2005.05.007"},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"L. Qi and Z. Luo,\n                      Tensor analysis: Spectral theory and special tensors\n                      , SIAM, Philadelphia, 2017,https:\/\/doi.org\/10.1137\/1.9781611974751.","DOI":"10.1137\/1.9781611974751"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894276370"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"N. Touzi,\n                      Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE\n                      , Fields Instit. Monogr. 29, Springer, New York, 2013,https:\/\/doi.org\/10.1007\/978-1-4614-4286-8.","DOI":"10.1007\/978-1-4614-4286-8"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/130915339"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1196923","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:44Z","timestamp":1787327264000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1196923"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1196923"],"URL":"https:\/\/doi.org\/10.1137\/18m1196923","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}