Minmaxxi

Minmaxxi is a theoretical construct in optimization and decision theory describing a family of min–max problems indexed by a parameter xi. For a real-valued function f: X × Y × Xi → R, where X is the decision space of the minimizing player and Y that of the adversary, the value V(xi) = min_{x∈X} max_{y∈Y} f(x,y,xi) captures the best guaranteed outcome against the worst response, for a given xi. The parameter xi encodes external conditions, data samples, or environment states. In robust formulations one may take the worst-case over xi, yielding min_{x∈X} max_{y∈Y} sup_{xi∈Xi} f(x,y,xi). Some authors treat xi as part of the problem data that can vary across instances.

Relation to other concepts: When f is convex in x and concave in y for every xi, a saddle point may exist and standard duality results apply. If xi is fixed, minmaxxi becomes the ordinary min–max problem; if the roles of x and xi are interchanged, one obtains a max–min variant. Different definitions of xi lead to alternative formulations and interpretations.

Computational aspects: Algorithms include saddle-point methods, decomposition, and robust optimization techniques. Nonconvexity, high dimensionality, or discontinuities in f can obstruct convergence and existence of saddle points. Practical approaches often rely on scenario sampling, regularization, or relaxation.

Applications: minmaxxi appears in robust decision making, adversarial training of machine learning models, and game-theoretic models under uncertainty, where xi represents scenarios, market states, or environmental conditions.

See also: min–max, maximin, saddle point, robust optimization.