Lowerbounding

Lowerbounding refers to the act of establishing a lower bound for a quantity, function, or set. In mathematics, a lower bound of a nonempty set S in an ordered set is a value b such that b is less than or equal to every element of S. The concept is central to analysis and order theory, providing a baseline against which other values can be compared.

The greatest lower bound, or infimum, of S is the largest value that still serves as a lower bound. If S has a minimum element, that minimum is itself a lower bound and also the greatest lower bound. Not all sets possess a minimum, yet they may still have a finite infimum. When the infimum is not attained by any element of S, the set has a lower bound but no minimum.

In optimization and algorithm design, lowerbounding is the process of producing a bound on the optimal value from below. Such bounds are crucial for assessing the quality of solutions and for pruning search spaces in global optimization methods. A typical approach is to relax the original problem to a simpler or cheaper problem whose solution provides a guaranteed lower bound to the original objective.

Common techniques for lower bounds include linear programming relaxations for integer programming, dual bounds from optimization theory, Lagrangian relaxations, and problem-specific inequalities or surrogate models. The effectiveness of a lower bound is measured by its tightness: closer to the true optimum yields more powerful pruning and faster convergence.

See also: bound (mathematics), infimum, minimum, supremum, optimization, branch and bound.