{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:38:38Z","timestamp":1787337518148,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-AC02-06CH11357"],"award-info":[{"award-number":["DE-AC02-06CH11357"]}],"id":[{"id":"10.13039\/100000015","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>This paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set. This is accomplished using lexicographic directional derivatives, a recently developed tool in nonsmooth analysis based on Nesterov's lexicographic differentiation. A nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric KKT system. It is shown that the sufficient conditions for this implicit function theorem variant are implied by a KKT point satisfying the linear independence constraint qualification and strong second-order sufficiency. Mirroring the classical theory, the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution. Practically implementable algorithms are provided for calculating the nonsmooth sensitivity system's unique solution, which is then used to furnish B-subdifferential elements of the primal and dual variable solutions by solving a linear equation system. Consequently, the findings in this article are computationally relevant since dedicated nonsmooth equation-solving and optimization methods display attractive convergence properties when supplied with such generalized derivative elements. The results have potential applications in nonlinear model predictive control and problems involving dynamic systems with mathematical programs embedded. Extending the theoretical treatments here to sensitivity analysis theory of other mathematical programs is also anticipated.<\/jats:p>","DOI":"10.1137\/17m1120385","type":"journal-article","created":{"date-parts":[[2018,2,1]],"date-time":"2018-02-01T12:29:50Z","timestamp":1517488190000},"page":"272-301","source":"Crossref","is-referenced-by-count":26,"title":["Generalized Sensitivity Analysis of Nonlinear Programs"],"prefix":"10.1137","volume":"28","author":[{"given":"Peter","family":"Stechlinski","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kamil A.","family":"Khan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paul I.","family":"Barton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"P. I. Barton, K. A. Khan, P. Stechlinski, and H. A. J. Watson,\n                      Computationally relevant generalized derivatives: Theory, evaluation and applications\n                      , Optim. Methods Softw., to appear,doi:10.1080\/10556788.2017.1374385.","DOI":"10.1080\/10556788.2017.1374385"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"J. F. Bonnans and A. Shapiro,\n                      Perturbation Analysis of Optimization Problems\n                      , Springer, New York, 2000.","DOI":"10.1007\/978-1-4612-1394-9"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"F. H. Clarke,\n                      Optimization and Nonsmooth Analysis\n                      , Classics in Appl. Math., SIAM, Philadelphia, 1990.","DOI":"10.1137\/1.9781611971309"},{"key":"atypb4","unstructured":"F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski,\n                      Nonsmooth Analysis and Control Theory\n                      , Grad. Texts in Math. 178, Springer, New York, 2008."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1109\/MCS.2008.919306"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581237"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0676-6"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"F. Facchinei and J.S. Pang,\n                      Finite-Dimensional Variational Inequalities and Complementarity Problems: Volume\n                      I, Springer, New York, 2003.","DOI":"10.1007\/b97543"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"F. Facchinei and J.S. Pang,\n                      Finite-Dimensional Variational Inequalities and Complementarity Problems: Volume\n                      II, Springer, New York, 2003.","DOI":"10.1007\/b97543"},{"key":"atypb10","unstructured":"A. V. Fiacco,\n                      Introduction to Sensitivity and Stability Analysis in Nonlinear Programming\n                      , Academic Press, New York, 1983."},{"key":"atypb11","unstructured":"A. V. Fiacco and G. P. McCormick,\n                      Nonlinear Programming: Sequential Unconstrained Minimization Techniques\n                      , Wiley, New York, 1968."},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"T. Gal,\n                      Postoptimal Analyses, Parametric Programming and Related Topics\n                      , 2nd ed., Walter de Gruyter, Berlin, 1995.","DOI":"10.1515\/9783110871203"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1080\/10556788.2013.796683"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"A. Griewank and A. Walther,\n                      Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation\n                      , 2nd ed., SIAM, Philadelphia, 2008.","DOI":"10.1137\/1.9780898717761"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2015.10.026"},{"key":"atypb17","first-page":"110","author":"Janin R.","year":"1984","journal-title":"New York"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"K. A. Khan,\n                      Branch-locking AD techniques for nonsmooth composite functions and nonsmooth implicit functions\n                      , Optim. Methods Softw., to appear,doi:10.1080\/10556788.2017.1341506.","DOI":"10.1080\/10556788.2017.1341506"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-014-0539-1"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"K. A. Khan and P. I. Barton,\n                      Generalized gradient elements for nonsmooth optimal control problems\n                      , in 53rd IEEE Conference on Decision and Control, 2014, pp. 1887-1892.","DOI":"10.1109\/CDC.2014.7039673"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1080\/10556788.2015.1025400"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2016.2644759"},{"key":"atypb23","unstructured":"D. Klatte and B. Kummer,\n                      Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications\n                      , Kluwer, Dordecht, the Netherlands, 2002."},{"key":"atypb24","first-page":"93","author":"Kojima M.","year":"1980","journal-title":"New York"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1287\/moor.15.2.286"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"C. Lemare\u0301chal, J. J. Strodiot, and A. Bihain,\n                      On a bundle algorithm for nonsmooth optimization\n                      , in Nonlinear Programming 4, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds., Academic Press, New York, 1981.","DOI":"10.1016\/B978-0-12-468662-5.50015-X"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-003-0452-0"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1994.1144"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"B. S. Mordukhovich,\n                      Variational Analysis and Generalized Differentiation I: Basic Theory\n                      , Springer, Berlin, 2006.","DOI":"10.1007\/3-540-31247-1"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-005-0633-0"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1287\/moor.21.2.401"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581275"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585934"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/BF02614400"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1287\/moor.5.1.43"},{"key":"atypb36","unstructured":"V. Roshchina,\n                      Two applications of lexicographic differentiation\n                      , presented at CIAO Showcase and Workshop, 2017,http:\/\/www.roshchina.com\/wp-content\/uploads\/2017\/04\/lexicographic.pdf."},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"S. Scholtes,\n                      Introduction to Piecewise Differentiable Equations\n                      , Springer, New York, 2012.","DOI":"10.1007\/978-1-4614-4340-7"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/0326037"},{"key":"atypb39","doi-asserted-by":"crossref","unstructured":"P. G. Stechlinski and P. I. Barton,\n                      Generalized derivatives of optimal control problems with nonsmooth differential-algebraic equations embedded\n                      , in Proceedings of the 55th IEEE Conference on Decision and Control, 2016, pp. 592-597.","DOI":"10.1109\/CDC.2016.7798333"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2016.10.041"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008669226453"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/BF00933296"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1021\/acs.iecr.7b03232"},{"key":"atypb44","unstructured":"J. H. Wilkinson,\n                      Rounding Errors in Algebraic Processes\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1963."},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1016\/j.jprocont.2016.05.002"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1120385","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:15:09Z","timestamp":1787336109000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1120385"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1120385"],"URL":"https:\/\/doi.org\/10.1137\/17m1120385","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}