{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T04:02:19Z","timestamp":1787371339350,"version":"build-2736575974"},"reference-count":6,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We consider a smooth finite dimensional parametric optimization problem ${\\cal P}(y)$ with objective function $f(x,y)$. Here, x and y denote the state variable and the parameter, respectively. In the case that $\\overline{x}$ is a strongly stable Karush--Kuhn--Tucker point for ${\\cal P}(\\overline{y})$, a neighborhood of $\\overline{x}$ contains a unique Karush--Kuhn--Tucker point $x(y)$ for ${\\cal P}(y)$, provided that y is sufficiently close to $\\overline{y}$. This gives rise to the critical value function $y\\mapsto\\varphi(y):=f(x(y),y)$. Under the additional assumption that the Mangasarian--Fromovitz constraint qualification is satisfied at $\\overline{x}$, we show that $\\varphi$ has finite modulus of concavity. That means $\\varphi$ becomes convex in a neighborhood of $\\overline{y}$ by adding to it the function $y\\mapsto (\\alpha\/2)\\cdot\\|y-\\overline{y}\\|^2$ for some $\\alpha&gt;0$. Moreover, we present an explicit upper bound for the $\\alpha$ to be used. The latter bound turns out to be sharp for problem data in general position.<\/jats:p>","DOI":"10.1137\/s1052623403434735","type":"journal-article","created":{"date-parts":[[2006,2,4]],"date-time":"2006-02-04T21:00:21Z","timestamp":1139086821000},"page":"1044-1053","source":"Crossref","is-referenced-by-count":2,"title":["Critical Value Functions have Finite Modulus of Concavity"],"prefix":"10.1137","volume":"16","author":[{"given":"Harald","family":"G\u00fcnzel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francisco Guerra","family":"Vazquez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hubertus Th.","family":"Jongen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,28]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01593777"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.21.4.783"},{"key":"R3","unstructured":"H. G\u00fcnzel and H. Th. Jongen,\n                      Strong Stability Implies Mangasarian\u2010Fromovitz Constraint Qualification\n                      , SIAM J. Optim., to appear."},{"key":"R4","volume-title":"Optimization theory","author":"Jongen Hubertus","year":"2004"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90028-0"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"MasakazuKojima, Strongly stable stationary solutions in nonlinear programs, Publ. Math. Res. Center Univ. Wisconsin, Vol. 43, Academic Press, New York, 1980, 93\u201313883a:90147","DOI":"10.1016\/B978-0-12-590240-3.50009-4"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1052623403434735","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:32:17Z","timestamp":1787333537000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1052623403434735"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/S1052623403434735"],"URL":"https:\/\/doi.org\/10.1137\/s1052623403434735","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}