Constraints

Optimization · Axiom Academy

Understanding how constraints shape and limit the feasible region in optimization problems Constraints define the feasible region - the set of all points where our solution is valid. Without constraints, the search space is unbounded. With constraints, we restrict our search to a specific region. Equality constraints require that a function equals a specific value, typically zero: g(x) = 0 . These constraints force solutions to lie on a specific surface or curve in the search space. Inequality constraints define one-sided boundaries: h(x) ≤ 0 or h(x) ≥ 0 . They create half-spaces that include all points on one side of a boundary. Box constraints are simple bounds on individual variables: a ≤ x ≤ b . They're the most common type of constraint and define rectangular (box-shaped) feasible regions. 5. Combining Multiple Constraints Real problems typically involve multiple constraints of different types. The feasible region is the intersection of all individual constraint regions. Optimizers must find solutions that satisfy all constraints simultaneously.

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