{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:26:43Z","timestamp":1787318803223,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>To minimize or upper-bound the value of a function \u201crobustly,\u201d we might instead minimize or upper-bound the \u201c$\\epsilon$-robust regularization,\u201d defined as the map from a point to the maximum value of the function within an $\\epsilon$-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius of a matrix leads to pseudospectral computations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small $\\epsilon&gt;0$, even if the original function is non-Lipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such functions have graphs that are finite unions of sets defined by finitely many polynomial inequalities, and are commonly encountered in applications.<\/jats:p>","DOI":"10.1137\/08073682x","type":"journal-article","created":{"date-parts":[[2009,12,11]],"date-time":"2009-12-11T18:17:06Z","timestamp":1260555426000},"page":"3080-3104","source":"Crossref","is-referenced-by-count":27,"title":["Lipschitz Behavior of the Robust Regularization"],"prefix":"10.1137","volume":"48","author":[{"given":"Adrian S.","family":"Lewis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C. H. Jeffrey","family":"Pang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,12,11]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. Ben-Tal and A. Nemirovski,\n                      Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications\n                      , SIAM, Philadelphia, 2001.","DOI":"10.1137\/1.9780898718829"},{"key":"R2","unstructured":"R. Benedetti and J.J. Risler,\n                      Real Algebraic and Semi-Algebraic Sets\n                      , Hermann, Paris, 1990."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,\n                      Linear Matrix Inequalities in System and Control Theory\n                      , Stud. Appl. 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