Inequality constraints

Objective. Interpret complementary slackness and KKT conditions.

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Learn: the key idea

Inequality constraints extend optimisation beyond equalities by allowing feasible regions bounded by curves like g(x)≤0, requiring the Karush–Kuhn–Tucker (KKT) conditions. At an optimum, ∇f + Σμᵢ∇gᵢ = 0 with μᵢ ≥ 0, gᵢ ≤ 0, and complementary slackness μᵢgᵢ = 0, meaning a multiplier is zero unless its constraint is active (binding). Example: minimise f(x)=x² subject to x≥1, i.e. g(x)=1−x≤0; since the unconstrained minimum x=0 is infeasible, the constraint binds at x=1 with μ>0, giving minimum value 1. Complementary slackness tells us exactly which constraints matter at the solution.

Worked example

Using KKT inequality-constraint reasoning, minimise f(x) = (x−3)² subject to x ≥ 5.

1 $Unconstrained minimum of f is at x = 3, which violates x \geq 5$
2 $So the constraint x \geq 5 is active, giving g(x) = 5 - x = 0, so x = 5$
3 $Check complementary slackness: μ > 0 is consistent since the constraint binds$
4 $Evaluate f(5) = (5-3)² = 4$

Answer: 4

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Common mistake. Students assign a positive multiplier to every constraint listed, ignoring complementary slackness; if a constraint is not active at the optimum its multiplier must be zero, not merely small.

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