In vector optimization, one investigates optimization problems with a vector-valued objective function. Fundamental works concerning vector optimization date back to F. Y. Edgeworth (1881) und V. Pareto (1896).

Some recommended books:

G. Eichfelder : Variable Ordering Structures in Vector Optimization,
Springer, 2014

A. Löhne : Vector Optimization with Infimum and Supremum, Springer, 2011

J. Jahn : Vector Optimization – Theory, Applications, and Extensions, Springer, Berlin, 2nd Edition, 2011

A. Göpfert, H. Riahi, Chr. Tammer and C. Zălinescu : Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 17, Springer-Verlag, New York, 2003