Minmaxmin

Minmaxmin is a term encountered in optimization and game theory, used to refer to a triple-nested optimization operation in which the order of min and max is specified and applied sequentially. Although not a widely standardized notation, it commonly describes problems of the form min over x of max over y of min over z of a function f(x, y, z), with the exact variable order depending on the context.

Formal definition and variants

For a function f defined on domains X, Y, Z, minmaxmin can denote min_{x in X} max_{y in Y} min_{z in Z} f(x, y, z). Other orderings exist, such as min_{x} min_{y} max_{z} f(x, y, z) or max-min-max sequences, and the meaning is sensitive to the problem’s interpretation and constraints. The triple-nested structure generalizes the familiar two-player minimax setting and often arises in hierarchical or sequential decision problems.

Theoretical considerations

Existence and characterization of optimal solutions for minmaxmin depend on properties like convexity, concavity, and compactness of the domains, as well as continuity of f. In two-player zero-sum settings, minimax theorems guarantee saddle points under suitable conditions; for triple-nested problems, analogous results are more delicate and may require specialized assumptions or reformulations.

Computational aspects

Solving minmaxmin typically involves nested optimization techniques, such as alternating minimization and maximization, bilevel programming, or problem reformulations using duality, KKT conditions, or epigraph methods. The computational burden grows with the number of nested levels and the dimensionality of the variables.

Applications and context

Minmaxmin appears in hierarchical decision making, robust optimization with multiple layers of adversarial or uncertain components, and multi-agent systems where sequential or nested worst-case objectives are relevant.

See also

Minimax, maximin, saddle point, bilevel optimization, nested optimization, game theory.