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We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.<\/jats:p>","DOI":"10.1007\/s10957-024-02528-w","type":"journal-article","created":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T03:09:35Z","timestamp":1727060975000},"page":"676-713","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0893-5604","authenticated-orcid":false,"given":"Thai Doan","family":"Chuong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinghuo","family":"Yu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chen","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrew","family":"Eberhard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chaojie","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,23]]},"reference":[{"key":"2528_CR1","doi-asserted-by":"crossref","unstructured":"Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization-Analysis, Algorithms and Engineering Applications, Philadelphia (2001)","DOI":"10.1137\/1.9780898718829"},{"issue":"2","key":"2528_CR2","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s10107-003-0454-y","volume":"99","author":"A Ben-Tal","year":"2004","unstructured":"Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. 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