KKT Condition Examples

Optimization · Axiom Academy

Master the Karush-Kuhn-Tucker optimality conditions through worked examples Example 1: Single Inequality Constraint Excellent work! You've mastered KKT conditions through three complete examples. Here's what we learned: The Four KKT Conditions: (1) Stationarity: ∇f(x*) + Σμᵢ∇gᵢ(x*) = 0, (2) Primal feasibility: gᵢ(x*) ≤ 0, (3) Dual feasibility: μᵢ ≥ 0, (4) Complementary slackness: μᵢgᵢ(x*) = 0 Active vs Inactive Constraints: A constraint is active when gᵢ(x*) = 0 (binding) and inactive when gᵢ(x*) 0 due to complementary slackness. Testing Different Cases: When solving KKT conditions, systematically test cases where constraints are either active (μᵢ > 0, gᵢ = 0) or inactive (μᵢ = 0, gᵢ < 0), then verify which case satisfies all conditions. Inactive Constraint Recognition: When a constraint is not active at the optimum, its multiplier must be zero, and the solution reduces to the unconstrained optimality condition ∇f(x*) = 0. Multiple Constraints: With both equality and inequality constraints, combine them systematically: use λ for equality constraints (no sign restriction) and μ ≥ 0 for inequality constraints. These conditions are necessary for optimality in constrained optimization. Practice identifying active constraints and applying complementary slackness to master KKT conditions!

This is the written version of the interactive lesson above. See the full Optimization course.