Equality Constraints

Optimization · Axiom Academy

LESSON Equality Constraints in Optimization Understanding how equality constraints restrict feasible regions and deriving optimality conditions using the Lagrangian method 1. Equality-Constrained Optimization Problem The general form of an equality-constrained optimization problem involves minimizing an objective function subject to equality constraints: Here, f: ℝⁿ → ℝ is the objective function, and h: ℝⁿ → ℝᵐ represents m equality constraints (with m < n). The constraints define an (n-m)-dimensional manifold in the n-dimensional space. 2. Reduction of Degrees of Freedom Each independent equality constraint reduces the problem's degrees of freedom by one. Starting with n variables and imposing m independent constraints, we have (n - m) remaining degrees of freedom. For example, the constraint x² + y² = 1 in ℝ² reduces the feasible region from a 2D plane to a 1D circle. At any feasible point, we can only move along the constraint surface. 3. Linear Independence Constraint Qualification (LICQ) For the optimality conditions to hold, we require a constraint qualification . The most common is the Linear Independence Constraint Qualification (LICQ): LICQ ensures that the constraints are truly independent and that the tangent space to the constraint manifold is well-defined. Without LICQ, the optimality conditions may fail.

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