{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:38:37Z","timestamp":1740155917035,"version":"3.37.3"},"reference-count":28,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Operations Research"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>We focus on some convex separable optimization problems, considered by the author in previous papers, for which problems, necessary and sufficient conditions or sufficient conditions have been proved, and convergent algorithms of polynomial computational complexity have been proposed for solving these problems. The concepts of well-posedness of optimization problems in the sense of Tychonov, Hadamard, and in a generalized sense, as well as calmness in the sense of Clarke, are discussed. It is shown that the convex separable optimization problems under consideration are calm in the sense of Clarke. The concept of stability of the set of saddle points of the Lagrangian in the sense of Gol'shtein is also discussed, and it is shown that this set is not stable for the \u201cclassical\u201d Lagrangian. However, it turns out that despite this instability, due to the specificity of the approach, suggested by the author for solving problems under consideration, it is not necessary to use modified Lagrangians but only the \u201cclassical\u201d Lagrangians. Also, a primal-dual analysis for problems under consideration in view of methods for solving them is presented.<\/jats:p>","DOI":"10.1155\/2013\/279030","type":"journal-article","created":{"date-parts":[[2013,4,15]],"date-time":"2013-04-15T17:00:32Z","timestamp":1366045232000},"page":"1-10","source":"Crossref","is-referenced-by-count":8,"title":["Well-Posedness and Primal-Dual Analysis of Some Convex Separable Optimization Problems"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2402-1554","authenticated-orcid":true,"given":"Stefan M.","family":"Stefanov","sequence":"first","affiliation":[{"name":"Department of Informatics, South-West University \u201cNeofit Rilski\u201d, 2700 Blagoevgrad, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"13","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008739510750"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1155\/IJMMS.2005.1339"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971309"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.2307\/2319313"},{"issue":"4","key":"6","first-page":"569","volume":"8","year":"1972","journal-title":"Economics and Mathematical Methods"},{"year":"1989","key":"10"},{"issue":"4","key":"27","first-page":"631","volume":"6","year":"1966","journal-title":"USSR Computational Mathematics and Mathematical Physics"},{"key":"1","series-title":"Lecture Notes in Pure and Applied Mathematics","volume-title":"Well-posed saddle point problems","volume":"86","year":"1983"},{"volume":"1543","year":"1993","series-title":"Lecture Notes in Mathematics","key":"5"},{"key":"8","first-page":"49","volume":"13","year":"1902","journal-title":"Bulletin Princeton University"},{"year":"1953","key":"9"},{"issue":"5","key":"21","first-page":"195","volume":"39","year":"1943","journal-title":"Doklady Akademii Nauk SSSR"},{"issue":"3","key":"22","first-page":"501","volume":"151","year":"1963","journal-title":"Doklady Akademii Nauk SSSR"},{"issue":"1","key":"23","first-page":"49","volume":"153","year":"1963","journal-title":"Doklady Akademii Nauk SSSR"},{"issue":"4","key":"24","first-page":"763","volume":"162","year":"1965","journal-title":"Doklady Akademii Nauk SSSR"},{"issue":"3","key":"25","first-page":"507","volume":"164","year":"1965","journal-title":"Doklady Akademii Nauk SSSR"},{"issue":"1","key":"26","first-page":"81","volume":"6","year":"1966","journal-title":"USSR Computational Mathematics and Mathematical Physics"},{"year":"1977","key":"28"},{"year":"1986","key":"29"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1.2.165"},{"issue":"3","key":"7","first-page":"568","volume":"10","year":"1974","journal-title":"Economics and Mathematical Methods"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171203210516"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1155\/S168712000402009X"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1155\/S1110757X04309071"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1155\/JAMDS\/2006\/89307"},{"issue":"4","key":"19","volume":"2006","year":"2006","journal-title":"Applied Mathematics Research eXpress"},{"issue":"5","key":"20","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1080\/09720502.2010.10700719","volume":"13","year":"2010","journal-title":"Journal of Interdisciplinary Mathematics"}],"container-title":["Advances in Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2013\/279030.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2013\/279030.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2013\/279030.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,21]],"date-time":"2017-06-21T07:22:31Z","timestamp":1498029751000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/aor\/2013\/279030\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":28,"alternative-id":["279030","279030"],"URL":"https:\/\/doi.org\/10.1155\/2013\/279030","relation":{},"ISSN":["1687-9147","1687-9155"],"issn-type":[{"type":"print","value":"1687-9147"},{"type":"electronic","value":"1687-9155"}],"subject":[],"published":{"date-parts":[[2013]]}}}