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One motivation to study StQPs under such sparsity restrictions is the high-dimensional portfolio selection problem with too many assets to handle, in particular, in the presence of transaction costs. Here, relying on modern conic optimization techniques, we present tractable convex relaxations for this relevant but difficult problem. We propose novel equivalent reformulations of these relaxations with significant dimensional reduction, which is essential for the tractability of these relaxations when the problem size grows. Moreover, we propose an instance generation procedure which systematically avoids too easy instances. Our extensive computational results illustrate the high quality of the relaxation bounds in a significant number of instances. Furthermore, in contrast with exact mixed-integer quadratic programming models, the solution time of the relaxations is very robust to the choices of the problem parameters. In particular, the reduced formulations achieve significant improvements in terms of the solution time over their counterparts.<\/jats:p>","DOI":"10.1007\/s12532-025-00285-z","type":"journal-article","created":{"date-parts":[[2025,7,11]],"date-time":"2025-07-11T09:45:35Z","timestamp":1752227135000},"page":"617-651","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Tighter yet more tractable relaxations and nontrivial instance generation for sparse standard quadratic optimization"],"prefix":"10.1007","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6288-9226","authenticated-orcid":false,"given":"Immanuel M.","family":"Bomze","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2650-0295","authenticated-orcid":false,"given":"Bo","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuzhou","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4141-3189","authenticated-orcid":false,"given":"E. 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