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When the underlying constraint set is symmetric, it is known that sparse projection can be efficiently computed via a sorting function. Motivated by the observation that some class of underlying constraint sets is partially symmetric, i.e., symmetric with respect to a certain index subset, this paper introduces the notion of semi-symmetry and develops conditions of sparse projection onto a semi-symmetric set. These conditions are characterized by a sorting function and certain combinatorial conditions. When the sparsity level or the number of multi-semi-symmetric index subsets is small or moderate, these conditions can be efficiently verified. Stationary point conditions are obtained for semi-symmetric sets, and the convergence of the projected gradient descent scheme with an improved step length is established in a general setting and is applied to sparse optimization. The proposed semi-symmetric sparse projection based schemes are tested on the joint sparsity and group sparsity problem from statistics and two finance problems subject to semi-symmetric constraints; numerical results demonstrate improved performance in terms of numerical accuracy and computation time.<\/jats:p>","DOI":"10.1007\/s10898-026-01592-y","type":"journal-article","created":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T01:31:41Z","timestamp":1770255101000},"page":"809-844","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sparse Projection onto Semi-Symmetric Sets with Applications to Sparse Optimization"],"prefix":"10.1007","volume":"94","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2172-4182","authenticated-orcid":false,"given":"Jinglai","family":"Shen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saeed","family":"Damadi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,5]]},"reference":[{"issue":"2","key":"1592_CR1","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-\u0141ojasiewicz inequality. 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