Complementary Slackness

For every inequality constraint $g_j(x) \le 0$ with multiplier $\mu_j \ge 0$: $\mu_j \cdot g_j(x) = 0$ → at least one of the two is zero:

• constraint non-binding ($g_j < 0$, room left) → $\mu_j = 0$ (relaxing it changes nothing)
• $\mu_j > 0$ → constraint binding ($g_j = 0$, full)

Example: B runs 200 of 500 MW → capacity non-binding → its Shadow Price = 0. A runs 100 of 100 MW → binding → −10.

Use by hand (active set): guess which constraints are binding → binding: $g_j = 0$, others: $\mu_j = 0$ → solve the equations → check $g_j \le 0$ and $\mu_j \ge 0$ → if violated, change the guess.

[Draft part- W7 notation] : $0 \le \mu_j \perp -g_j(x) \ge 0$ (both ≥ 0, product = 0) → Mixed Complementarity Problem.

Links: KKT Conditions, Shadow Price, Duality