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Design issues are relatively much less discussed for hierarchical linear models but there is an increasing interest as these models grow in popularity. This paper discusses the\n                    <jats:italic>G<\/jats:italic>\n                    -optimality for predicting individual parameters in such models and establishes an equivalence theorem for confirming the\n                    <jats:italic>G<\/jats:italic>\n                    -optimality of an approximate design. Because the criterion is non-differentiable and requires solving multiple nested optimization problems, it is much harder to find and study\n                    <jats:italic>G<\/jats:italic>\n                    -optimal designs analytically. We propose a nature-inspired meta-heuristic algorithm called competitive swarm optimizer (CSO) to generate\n                    <jats:italic>G<\/jats:italic>\n                    -optimal designs for linear mixed models with different means and covariance structures. We further demonstrate that CSO is flexible and generally effective for finding the widely used locally\n                    <jats:italic>D<\/jats:italic>\n                    -optimal designs for nonlinear models with multiple interacting factors and some of the random effects are correlated. Our numerical results for a few examples suggest that\n                    <jats:italic>G<\/jats:italic>\n                    and\n                    <jats:italic>D<\/jats:italic>\n                    -optimal designs may be equivalent and we establish that\n                    <jats:italic>D<\/jats:italic>\n                    and\n                    <jats:italic>G<\/jats:italic>\n                    -optimal designs for hierarchical linear models are equivalent when the models have only a random intercept only. The challenging mathematical question of whether their equivalence applies more generally to other hierarchical models remains elusive.\n                  <\/jats:p>","DOI":"10.1007\/s00500-021-06061-0","type":"journal-article","created":{"date-parts":[[2021,8,7]],"date-time":"2021-08-07T12:02:43Z","timestamp":1628337763000},"page":"13549-13565","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["G-optimal designs for hierarchical linear models: an equivalence theorem and a nature-inspired meta-heuristic algorithm"],"prefix":"10.1007","volume":"25","author":[{"given":"Xin","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"RongXian","family":"Yue","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zizhao","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weng Kee","family":"Wong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,8,7]]},"reference":[{"key":"6061_CR1","doi-asserted-by":"publisher","DOI":"10.1002\/0470857005","volume-title":"Applied optimal designs","author":"M Berger","year":"2005","unstructured":"Berger M, Wong WK (2005) Applied optimal designs. 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