Optimal matching means creating treated-versus-control matched pairs, sets, or blocks as the solution to an optimization problem. Commonly, two or more treated individuals may want the same control as their closest or best match; so, an optimal match must allocate controls in a way that produces the best match in some overall sense. As is often true, framing a practical problem as an optimization problem opens up a world of new tools and insights. For a practical introduction to this topic, I would start with the my open access review, Rosenbaum (2020) below; for more practical detail, I would look at Part II of my book Design of Observational Studies (Springer 2020). For a review of some of the theory, I would start with Rosenbaum and Zubizarreta (2023).
Rosenbaum PR. Optimal matching for observational studies. Journal of the American Statistical Association. 1989 Dec 1;84(408):1024-32. In JSTOR: www.jstor.org/stable/2290079 https://doi.org/10.2307/2290079
Rosenbaum PR. A characterization of optimal designs for observational studies. Journal of the Royal Statistical Society B. 1991 Jul;53(3):597-610. In JSTOR: www.jstor.org/stable/2345589 https://doi.org/10.1111/j.2517-6161.1991.tb01848.x
Rosenbaum PR, Ross RN, Silber JH. Minimum distance matched sampling with fine balance in an observational study of treatment for ovarian cancer. Journal of the American Statistical Association. 2007 Mar 1;102(477):75-83. In JSTOR: www.jstor.org/stable/27639821 https://doi.org/10.1198/016214506000001059
Zubizarreta JR, Reinke CE, Kelz RR, Silber JH, Rosenbaum PR. Matching for several sparse nominal variables in a case-control study of readmission following surgery. American Statistician. 2011 Nov 1;65(4):229-38. Public access via the US National Library of Medicine https://pmc.ncbi.nlm.nih.gov/articles/PMC4237023/pdf/nihms628224.pdf https://doi.org/10.1198/tas.2011.11072
Rosenbaum PR. Optimal matching of an optimally chosen subset in observational studies. Journal of Computational and Graphical Statistics. 2012 Jan 1;21(1):57-71. In JSTOR: www.jstor.org/stable/23248823 https://doi.org/10.1198/jcgs.2011.09219
Zubizarreta JR, Paredes RD, Rosenbaum PR. Matching for Balance, Pairing for Heterogeneity in an Observational Study of the Effectiveness of For-Profit and Not-For-Profit High Schools in Chile. Annals of Applied Statistics. 2014 Jan 1;8(361):204-31. Open access at ProjectEuclid: https://doi.org/10.1214/13-AOAS713
Pimentel SD, Kelz RR, Silber JH, Rosenbaum PR. Large, sparse optimal matching with refined covariate balance in an observational study of the health outcomes produced by new surgeons. Journal of the American Statistical Association. 2015 Apr 3;110(510):515-27. Public access via the US National Library of Medicine: https://pubmed.ncbi.nlm.nih.gov/26273117/ In JSTOR: www.jstor.org/stable/24739473 https://doi.org/10.1080/01621459.2014.997879
Yu R, Silber JH, Rosenbaum PR. Matching methods for observational studies derived from large administrative databases. Statistical Science. 2020 Aug 1;35(3):338-55. In JSTOR: www.jstor.org/stable/26997902 Open access in ProjectEuclid https://doi.org/10.1214/19-STS699
Rosenbaum PR. Modern algorithms for matching in observational studies. Annual Review of Statistics and Its Application. 2020 Mar 7;7(1):143-76. Open access: https://doi.org/10.1146/annurev-statistics-031219-041058
Zhang B, Small DS, Lasater KB, McHugh M, Silber JH, Rosenbaum PR. Matching one sample according to two criteria in observational studies. Journal of the American Statistical Association. 2023 Apr 3;118(542):1140-51. https://doi.org/10.1080/01621459.2021.1981337
Rosenbaum PR, Zubizarreta JR. Optimization techniques in multivariate matching. Handbook of Matching and Weighting Adjustments for Causal Inference. 2023 Apr 11:63-86.