Ibounds

Ibounds are a formal construct used in mathematics, computer science, and optimization to denote the bounds of indexed variables within a model. For each index i, the corresponding bound is the closed interval [l_i, u_i], which contains the feasible values of the variable x_i in all admissible solutions. Collectively, the family { [l_i, u_i] | i in I } is referred to as the Ibounds of the system.

In interval arithmetic and constraint propagation, Ibounds are propagated through equations and inequalities to refine feasible regions. When constraints relate variables, the bounds can be tightened by considering the relationships, leading to narrower intervals that still contain all feasible solutions. This refinement is useful in robust optimization, static analysis, and mathematical programming where exact enumeration is impractical.

Example: Consider a system with x1 ∈ [0,5] and x2 ∈ [2,7], constrained by x1 + x2 = 8. From x2 = 8 - x1 and x2 ∈ [2,7], we obtain x1 ∈ [1,6]. Intersect with the original x1 bound [0,5] to get x1 ∈ [1,5], which implies x2 = 8 - x1 ∈ [3,7]. The resulting Ibounds reflect the feasible region more accurately than the initial bounds alone.

Ibounds are used in various domains, including linear and nonlinear programming, SMT-based verification, and data validation, where knowing tight bounds supports pruning, risk assessment, and numerical stability. See also interval arithmetic; bound propagation; robust optimization.