minXmaxX

MinXmaxX is a term used in optimization and decision theory to denote a min-max problem: choosing a decision variable x to minimize the maximum value among a family of criteria or scenarios. The formal formulation is min_{x ∈ X} max_{i ∈ I} f_i(x), where X is a feasible set and {f_i} is a collection of real-valued functions. The inner maximum captures the worst-case performance across scenarios or criteria.

If each f_i is convex in x and X is convex, then F(x) = max_i f_i(x) is also convex, making the problem a convex optimization problem. When the functions are affine, the problem can often be reformulated as a linear program by introducing a scalar variable t that upper-bounds all f_i(x) (minimize t subject to f_i(x) ≤ t for all i).

Computational considerations vary with problem structure. In general, minXmaxX can be difficult if f_i are non-convex or the index set I is large. Techniques include subgradient methods for convex cases, scenario-based robust optimization, cutting-plane methods, and, in certain structured instances, semidefinite programming.

Applications of minXmaxX span robust design and decision making under uncertainty, supply chain risk management, control systems with disturbances, and machine learning approaches emphasizing adversarial robustness. The framework is also related to game-theoretic concepts such as saddle points and to duality in convex optimization.

See also: minimax, maximin, saddle point, robust optimization, convex optimization, linear programming, duality.