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Under a suitable regularity condition, an exact conic programming relaxation is established based on a positivity characterization of a max function over a conic convex system. Further, we consider a general conic minimax <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\rho $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c1<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-convex polynomial optimization problem, which is defined by appropriately extending the notion of conic convexity of a vector-valued mapping. For this problem, it is shown that a Karush-Kuhn-Tucker condition at a global minimizer is necessary and sufficient for ensuring an exact relaxation with attainment of the conic programming relaxation. The exact conic programming relaxations are applied to SOS-convex polynomial programs, where appropriate choices of the data allow the associated conic programming relaxation to be reformulated as a semidefinite programming problem. In this way, we can further elaborate the obtained results for other special settings including conic robust SOS-convex polynomial problems and difference of SOS-convex polynomial programs.<\/jats:p>","DOI":"10.1007\/s10898-025-01465-w","type":"journal-article","created":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T00:55:27Z","timestamp":1738025727000},"page":"743-763","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Conic relaxations for conic minimax convex polynomial programs with extensions and applications"],"prefix":"10.1007","volume":"91","author":[{"given":"Thai","family":"Doan Chuong","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7064-1239","authenticated-orcid":false,"given":"Jos\u00e9","family":"Vicente-P\u00e9rez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,28]]},"reference":[{"key":"1465_CR1","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s10107-011-0457-z","volume":"135","author":"AA Ahmadi","year":"2012","unstructured":"Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not SOS-convex. 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