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Inequality constraints are a fundamental concept in mathematical optimization. They are limitations on the variables in a problem that restrict their values to a certain range, typically expressed as a less than or equal to or greater than or equal to relationship. For example, in a production planning problem, an inequality constraint might represent the limited availability of raw materials.

These constraints are a key component of nonlinear programming problems, where the objective function or the constraints themselves are nonlinear. They differentiate optimization problems from those with only equality constraints, which require variables to take on specific values. The presence of inequality constraints often makes optimization problems more complex to solve, requiring specialized algorithms.

The set of feasible solutions to an optimization problem is defined by all points that satisfy all the given constraints, both equality and inequality. The optimal solution must lie within this feasible region. Understanding and correctly formulating inequality constraints is crucial for accurately modeling real-world problems and finding meaningful solutions. Many algorithms for solving nonlinear programming problems, such as gradient projection methods or sequential quadratic programming, are designed to handle these types of constraints effectively.