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Robust optimization can account for these uncertainties during treatment planning, for which the minimax approach optimizes the worst\u2010case plan quality.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Purpose<\/jats:title>\n                    <jats:p>This work will develop an efficient minimax robust optimization algorithm for improving plan quality and computational efficiency.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Methods<\/jats:title>\n                    <jats:p>The proposed method reformulates the minimax problem so that it can be conveniently solved by the first\u2010order operator\u2010splitting algorithm (OS). That is, the reformulated problem is split into several subproblems, which either admit a closed\u2010form solution or can be efficiently solved as a linear system.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Results<\/jats:title>\n                    <jats:p>\n                      The proposed method OS was demonstrated with improved plan quality, robustness, and computational efficiency, compared to robust optimization via stochastic programming (SP) and current minimax robust method via minimax stochastic programming (MSP). For example, in a prostate case, compared to MSP and SP, OS decreased the max target dose from 140% and 121% to 118%, and the mean femoral head dose from 28.6% and 26.3% to 24.8%. In terms of robustness, OS reduced the robustness variance (RV\n                      <jats:sub>120<\/jats:sub>\n                      ) of the target from 56.07 and 0.30 to 0.04. Compared to MSP, OS decreased the computational time from 16.4 min to 1.7 min.\n                    <\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Conclusions<\/jats:title>\n                    <jats:p>A novel operator\u2010splitting minimax robust optimization is proposed with improved plan quality and computational efficiency, compared to conventional minimax robust optimization method MSP and probabilistic robust optimization method SP.<\/jats:p>\n                  <\/jats:sec>","DOI":"10.1002\/mp.17929","type":"journal-article","created":{"date-parts":[[2025,6,8]],"date-time":"2025-06-08T10:12:11Z","timestamp":1749377531000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Efficient operator\u2010splitting minimax algorithm for robust 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