The author considers a range of problems related to differential calculus of set-valued maps. The proposed constructions are developed on the base of the concept of a multiaffine mapping or, simply, a multiaffine. By the definition a multiaffine is the mapping whose graph may be represented as the union of the graphs of a family of single-valued functions \(L(x) = Ax + b\), where \(x \in R^m\), \(A\) is an \(n \times m\) matrix and \(b \in R^n\). The family is characterized by a set \(C\) of pairs \((A,b)\) compact in the appropriate product of spaces. The set \(C\) is called a generator. The latter is not always defined by the multiaffine uniquely.
A set-valued analog of the differential, which is called directive, is introduced via approximation of a multifunction by a multiaffine. Directional, partial and one-sided directional are defined. Basic rules of the calculus of directives are derived. Using these rules, necessary conditions of optimality in min-max and max-min problems are obtained.
The author provides an extensive survey and comparison with other works in this direction. It is erroneously stated that an eclipset [see C. Lemaréchal and J. Zowe, Optimization 22, No. 1, 3-37 (1991; Zbl 0743.47039)] can be represented as a multiaffine. An alternative approach to extend the concept of differential to set-valued maps is described in a reviewer’s article [Dokl. Akad. Nauk, Ross. Akad. Nauk 340, 164-167 (1995)] which is not included in the survey.
In the last sections of the paper the space of multiaffines is supplied with metrics, that allows to investigate set-valued differential equations. Existence and uniqueness theorems are proved.