Constraints

Constraints are conditions or limits that restrict the set of possible solutions or actions within a system. They may be physical, mathematical, contractual, or normative, and they typically define what is feasible rather than what is ideal. By constraining choices, constraints shape behavior, outcomes, and design.

In mathematics and optimization, a constraint is a condition that must be satisfied by the decision variables. Constraints can be equalities or inequalities. An optimization problem seeks to optimize an objective function subject to these constraints, producing a feasible region. Linear programming uses linear constraints; nonlinear programming handles nonlinear ones; integer programming restricts variables to discrete values. In constraint satisfaction problems, constraints define allowable combinations and constraint propagation methods reduce variable domains to prune the search space.

In computer science, constraint programming is a paradigm where problems are modeled by variables with finite domains and a set of constraints. Solutions are assignments that satisfy all constraints. This approach is applied to scheduling, configuration, resource allocation, and timetabling. In databases, integrity constraints enforce data validity, such as domain restrictions and relational constraints like primary keys and foreign keys.

Constraints vary in rigidity. Hard constraints must be satisfied for feasibility; soft constraints reflect preferences and may be violated with a cost. External constraints arise from laws, standards, and external resources, while internal constraints come from system design or user requirements. Effective constraint management involves identifying, representing, and, when possible, relaxing or prioritizing constraints to obtain acceptable solutions.