Inequality Constraints

Optimization · Axiom Academy

Understanding how inequality constraints shape feasible regions and optimal solutions in constrained optimization 1. Inequality-Constrained Optimization An inequality-constrained optimization problem seeks to minimize an objective function subject to constraints that create boundaries rather than surfaces. The constraint g(x) ≤ 0 creates a half-space: all points where g(x) is non-positive are feasible (shown in blue). The boundary occurs where g(x) = 0. Unlike equality constraints that restrict solutions to an exact surface, inequality constraints create a feasible region with an interior and a boundary. The feasible region (blue) includes all interior points where g(x) 0. 3. Active and Inactive Constraints At any feasible point, a constraint is either active (binding) or inactive (non-binding), which determines whether it influences the local geometry. At point A, the constraint is inactive (slack). At point B, it is active (binding). Only at B does the constraint restrict movement. 4. Complementary Slackness Condition The complementary slackness condition captures the relationship between constraint activity and Lagrange multipliers in the KKT conditions. If a constraint is inactive (gᵢ 0), the constraint must be active (gᵢ = 0). 5. Boundary vs Interior Optima Optimal solutions can occur either at the boundary (with active constraints) or in the interior (with all constraints inactive).

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