{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T19:08:24Z","timestamp":1775070504170,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1-3","license":[{"start":{"date-parts":[[2021,11,13]],"date-time":"2021-11-13T00:00:00Z","timestamp":1636761600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,11,13]],"date-time":"2021-11-13T00:00:00Z","timestamp":1636761600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2022,6]]},"DOI":"10.1007\/s10957-021-01959-z","type":"journal-article","created":{"date-parts":[[2021,11,13]],"date-time":"2021-11-13T16:02:17Z","timestamp":1636819337000},"page":"574-597","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Two Optimal Value Functions in Parametric Conic Linear Programming"],"prefix":"10.1007","volume":"193","author":[{"given":"Nguyen Ngoc","family":"Luan","sequence":"first","affiliation":[]},{"given":"Do Sang","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Nguyen Dong","family":"Yen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,11,13]]},"reference":[{"key":"1959_CR1","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/BF01758841","volume":"8","author":"I Adler","year":"1992","unstructured":"Adler, I., Monteiro, R.D.C.: A geometric view of parametric linear programming. Algorithmica 8, 161\u2013176 (1992)","journal-title":"Algorithmica"},{"key":"1959_CR2","unstructured":"An, D.T.V.: Subdifferentials of Optimal Value Functions in Parametric Convex Optimization Problems. Ph.D. Dissertation, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi (2018)"},{"key":"1959_CR3","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1080\/00036811.2014.890710","volume":"94","author":"DTV An","year":"2015","unstructured":"An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal. 94, 108\u2013128 (2015)","journal-title":"Appl. Anal."},{"key":"1959_CR4","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/BFb0120841","volume":"10","author":"A Auslender","year":"1979","unstructured":"Auslender, A.: Differential stability in nonconvex and nondifferentiable programming. Math. Program. Study 10, 29\u201341 (1979)","journal-title":"Math. Program. Study"},{"key":"1959_CR5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718829","volume-title":"Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications","author":"A Ben-Tal","year":"2001","unstructured":"Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, Philadelphia (2001)"},{"key":"1959_CR6","doi-asserted-by":"crossref","unstructured":"Berkelaar, A.B., Roos, K., Terlaky, T.: The optimal set and optimal partition approach. In: Gal, T., Greenberg, H.J. (eds.) Advances in Sensitivity Analysis and Parametric Programming, pp. 6.1-6.45. Kluwer Academic, Dordrecht (1997)","DOI":"10.1007\/978-1-4615-6103-3_6"},{"key":"1959_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1394-9","volume-title":"Perturbation Analysis of Optimization Problems","author":"JF Bonnans","year":"2000","unstructured":"Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)"},{"key":"1959_CR8","unstructured":"Cheung, Y.-L.: Preprocessing and Reduction for Semidefinite Programming via Facial Reduction: Theory and Practice. Ph.D. Dissertation, Department of Combinatorics & Optimization, University of Waterloo (2013)"},{"key":"1959_CR9","unstructured":"Cheung, Y.-L., Wolkowicz, H.: Sensitivity analysis of semidefinite programs without strong duality, tech. rep., (2014). (http:\/\/www.optimization-online.org\/DB_FILE\/2014\/06\/4416.pdf)"},{"key":"1959_CR10","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1016\/j.orl.2017.03.003","volume":"45","author":"TD Chuong","year":"2017","unstructured":"Chuong, T.D., Jeyakumar, V.: Convergent conic linear programming relaxations for cone-convex polynomial programs. Oper. Res. Lett. 45, 220\u2013226 (2017)","journal-title":"Oper. Res. Lett."},{"key":"1959_CR11","volume-title":"Introduction to Sensitivity and Stability Analysis in Nonlinear Programming","author":"AV Fiacco","year":"1983","unstructured":"Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)"},{"key":"1959_CR12","doi-asserted-by":"crossref","unstructured":"Gauvin, J.: Formulae for the sensitivity analysis of linear programming problems. In: Approximation. Optimization and Mathematical Economics, pp. 117\u2013120. Physica, Heidelberg (2001)","DOI":"10.1007\/978-3-642-57592-1_10"},{"key":"1959_CR13","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/BFb0120984","volume":"19","author":"J Gauvin","year":"1982","unstructured":"Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Study 19, 101\u2013119 (1982)","journal-title":"Math. Program. Study"},{"key":"1959_CR14","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BFb0121211","volume":"21","author":"J Gauvin","year":"1984","unstructured":"Gauvin, J., Dubeau, F.: Some examples and counterexamples for the stability analysis of nonlinear programming problems. Math. Program. Study 21, 69\u201378 (1984)","journal-title":"Math. Program. Study"},{"key":"1959_CR15","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1137\/0315020","volume":"15","author":"J Gauvin","year":"1977","unstructured":"Gauvin, J., Tolle, J.W.: Differential stability in nonlinear programming. SIAM J. Control Optim. 15, 294\u2013311 (1977)","journal-title":"SIAM J. Control Optim."},{"key":"1959_CR16","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1016\/S0168-9274(98)00102-0","volume":"29","author":"D Goldfarb","year":"1999","unstructured":"Goldfarb, D., Scheinberg, K.: On parametric semidefinite programming. Appl. Numer. Math. 29, 361\u2013377 (1999)","journal-title":"Appl. Numer. Math."},{"key":"1959_CR17","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1287\/moor.9.2.208","volume":"9","author":"B Gollan","year":"1984","unstructured":"Gollan, B.: On the marginal function in nonlinear programming. Math. Oper. Res. 9, 208\u2013221 (1984)","journal-title":"Math. Oper. Res."},{"key":"1959_CR18","doi-asserted-by":"crossref","unstructured":"Hauenstein, J.D., Mohammad-Nezhad, A., Tang, T., Terlaky, T.: On computing the nonlinearity interval in parametric semidefinite optimization, (2019). arXiv:1908.10499","DOI":"10.1080\/02331934.2019.1671382"},{"key":"1959_CR19","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1080\/02331930701763405","volume":"58","author":"T Hoheisel","year":"2009","unstructured":"Hoheisel, T., Kanzow, C.: On the Abadie and Guignard constraint qualifications for mathematical programmes with vanishing constraints. Optimization 58, 431\u2013448 (2009)","journal-title":"Optimization"},{"key":"1959_CR20","unstructured":"Jansen, B., Roos, C., Terlaky, T.: An interior point approach to postoptimal and parametric analysis in linear programming. Report No. 92\u201321. Faculty of Technical Mathematics and Information\/Computer science, Delft University of Technology (1992)"},{"key":"1959_CR21","first-page":"1823","volume":"13","author":"GM Lee","year":"2009","unstructured":"Lee, G.M., Tam, N.N., Yen, N.D.: Stability of a class of quadratic programs with a conic constraint. Taiwanese. J. Math. 13, 1823\u20131836 (2009)","journal-title":"Taiwanese. J. Math."},{"key":"1959_CR22","doi-asserted-by":"crossref","unstructured":"Li, W.: Abadie\u2019s constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim. 7, 966\u2013978 (1997)","DOI":"10.1137\/S1052623495287927"},{"key":"1959_CR23","doi-asserted-by":"crossref","unstructured":"Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming. Internat. Ser. Oper. Res. Management Sci.\u00a0228, 4th ed., Springer, Cham (2016)","DOI":"10.1007\/978-3-319-18842-3_6"},{"key":"1959_CR24","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1080\/02331934.2019.1671382","volume":"69","author":"A Mohammad-Nezhad","year":"2020","unstructured":"Mohammad-Nezhad, A., Terlaky, T.: Parametric analysis of semidefinite optimization. Optimization 69, 187\u2013216 (2020)","journal-title":"Optimization"},{"key":"1959_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-31246-3","volume-title":"Variational Analysis and Generalized Differentiation, I: Basic Theory","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)"},{"key":"1959_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-31246-3","volume-title":"Variational Analysis and Generalized Differentiation, II: Applications","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)"},{"key":"1959_CR27","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s10107-007-0120-x","volume":"116","author":"BS Mordukhovich","year":"2009","unstructured":"Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B 116, 369\u2013396 (2009)","journal-title":"Math. Program. Ser. B"},{"key":"1959_CR28","doi-asserted-by":"publisher","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"1959_CR29","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970524","volume-title":"Conjugate Duality and Optimization. Regional Conference Series in Applied Mathematics","author":"RT Rockafellar","year":"1974","unstructured":"Rockafellar, R.T.: Conjugate Duality and Optimization. Regional Conference Series in Applied Mathematics, vol. 16. SIAM, Philadelphia (1974)"},{"key":"1959_CR30","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1007\/BFb0120958","volume":"17","author":"RT Rockafellar","year":"1982","unstructured":"Rockafellar, R.T.: Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Program. Study 17, 28\u201366 (1982)","journal-title":"Math. Program. Study"},{"key":"1959_CR31","doi-asserted-by":"publisher","DOI":"10.1515\/9781400841059","volume-title":"Nonlinear Optimization","author":"A Ruszczynski","year":"2006","unstructured":"Ruszczynski, A.: Nonlinear Optimization. Princeton University Press, New Jersey (2006)"},{"key":"1959_CR32","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1007\/s10957-020-01780-0","volume":"188","author":"Y Sekiguchi","year":"2021","unstructured":"Sekiguchi, Y., Waki, H.: Perturbation analysis of singular semidefinite programs and its applications to control problems. J. Optim. Theory Appl. 188, 52\u201372 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"1959_CR33","first-page":"301","volume":"77","author":"A Shapiro","year":"1997","unstructured":"Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Program. Ser. B 77, 301\u2013320 (1997)","journal-title":"Math. Program. Ser. B"},{"key":"1959_CR34","doi-asserted-by":"crossref","unstructured":"Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.A., L\u00f3pez, M.A. (eds.) Semi-Infinite Programming: Recent Advances, pp. 135\u2013165, Springer (2001)","DOI":"10.1007\/978-1-4757-3403-4_7"},{"key":"1959_CR35","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/PL00011422","volume":"90","author":"JF Sturm","year":"2001","unstructured":"Sturm, J.F., Zhang, S.: On sensitivity of central solutions in semidefinite programming. Math. Program. 90, 205\u2013227 (2001)","journal-title":"Math. Program."},{"key":"1959_CR36","doi-asserted-by":"publisher","first-page":"1019","DOI":"10.1137\/0329056","volume":"29","author":"L Thibault","year":"1991","unstructured":"Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019\u20131036 (1991)","journal-title":"SIAM J. Control Optim."},{"key":"1959_CR37","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/s40306-015-0130-3","volume":"41","author":"NN Thieu","year":"2016","unstructured":"Thieu, N.N.: Some differential estimates in linear programming. Acta Math. Vietnam. 41, 243\u2013249 (2016)","journal-title":"Acta Math. Vietnam."},{"key":"1959_CR38","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1017\/S0962492901000071","volume":"10","author":"M Todd","year":"2001","unstructured":"Todd, M.: Semidefinite optimization. Acta Numer. 10, 515\u2013560 (2001)","journal-title":"Acta Numer."},{"key":"1959_CR39","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1023\/A:1022662120550","volume":"93","author":"ND Yen","year":"1997","unstructured":"Yen, N.D.: Stability of the solution set of perturbed nonsmooth inequality systems and application. J. Optim. Theory Appl. 93, 199\u2013225 (1997)","journal-title":"J. Optim. Theory Appl."},{"key":"1959_CR40","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1023\/B:JOTA.0000042528.76868.22","volume":"122","author":"EA Yildirim","year":"2004","unstructured":"Yildirim, E.A.: Unifying optimal partition approach to sensitivity analysis in conic optimization. J. Optim. Theory Appl. 122, 405\u2013423 (2004)","journal-title":"J. Optim. Theory Appl."},{"key":"1959_CR41","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/PL00011423","volume":"90","author":"EA Yildirim","year":"2001","unstructured":"Yildirim, E.A., Todd, M.: Sensitivity analysis in linear programming and semidefinite programming using interior-point methods. Math. Program. 90, 229\u2013261 (2001)","journal-title":"Math. Program."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-021-01959-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-021-01959-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-021-01959-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,14]],"date-time":"2022-05-14T20:30:33Z","timestamp":1652560233000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-021-01959-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,13]]},"references-count":41,"journal-issue":{"issue":"1-3","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["1959"],"URL":"https:\/\/doi.org\/10.1007\/s10957-021-01959-z","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,13]]},"assertion":[{"value":"10 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 October 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}