What do the KKT conditions require at the optimum of an inequality-constrained problem?

## The setup KKT - Karush-Kuhn-Tucker - generalizes Lagrange multipliers from equalities to inequalities. Fix the standard form: `minimize f(x) subject to g_i(x) <= 0 for i = 1..m, and h_j(x) = 0 for j = 1..p` Every inequality is written with zero on the right, and the Lagrangian carries one multiplier per constraint: `L(x, mu, lambda) = f(x) + sum_i mu_i * g_i(x) + sum_j lambda_j * h_j(x)` ## The four conditions **1. Stationarity.** The gradient of `L` with respect to `x` vanishes: `grad f + sum_i mu_i * grad g_i + sum_j lambda_j * grad h_j = 0`. The objective's downhill pull is exactly balanced by the constraints pushing back. **2. Primal feasibility.** `g_i(x) <= 0` for all `i` and `h_j(x) = 0` for all `j`. The point must actually be legal - stationarity of the Lagrangian alone does not guarantee this. **3. Dual feasibility.** `mu_i >= 0` for every inequality multiplier. Equality multipliers `lambda_j` are unrestricted in sign. **4. Complementary slackness.** `mu_i * g_i(x) = 0` for every `i`. All four must hold simultaneously; quoting only stationarity is the classic incomplete answer. ## Reading complementary slackness The product `mu_i * g_i(x) = 0` forces a case split per constraint. Either `g_i(x) = 0` - the constraint is **active**, the optimum sits on its boundary, and its multiplier is allowed to be positive - or `g_i(x) < 0` - the constraint is **slack**, the optimum sits strictly inside the region it permits, and then `mu_i` must be zero. That second case is the intuitive one. Picture a bowl-shaped objective whose unconstrained minimum lies well inside the allowed region, plus a boundary far away from it. Nothing about that boundary changes where the minimum sits: delete the constraint and the same point is returned. Zero multiplier, zero shadow price, no influence. Now shrink the region until the boundary cuts through the bowl. The unconstrained minimum becomes illegal, the solution slides onto the boundary, the constraint becomes active, and its multiplier turns positive - it now has a price, because loosening it would let the objective fall further. This is also why constrained optimization is combinatorially awkward: solving the problem is essentially discovering which subset of inequalities is active, and there are exponentially many candidate subsets. Algorithms differ mainly in how they search that active set without enumerating it. ## Why inequality multipliers must be nonnegative An equality constraint pins you from both sides, so its multiplier may point either way and its sign is a convention. An inequality only resists motion in one direction - out of the feasible region. Under the sign convention above, stationarity says `grad f = -sum_i mu_i * grad g_i`, meaning the objective wants to move in the direction that would increase `g_i` past zero, and the constraint is what stops it. A negative `mu_i` would describe a constraint pulling you *toward* infeasibility, which is not what a constraint does; equivalently, it would say that tightening the constraint improves the objective, which is impossible since tightening only removes options. ## Necessary, sufficient, or neither KKT conditions are **necessary** at a local minimum only if a **constraint qualification** holds - a regularity condition ruling out degenerate constraint geometry. A common one is linear independence of the gradients of the active constraints at the point. Without such a condition, a genuine minimum can fail every KKT test, so failing KKT is not by itself proof that a point is suboptimal. They become **sufficient**, and sufficient for a *global* minimum, when the problem is convex: `f` convex, the `g_i` convex, the `h_j` affine, plus a qualification such as Slater's condition (some strictly feasible point exists). In that setting any point satisfying KKT is a global optimum, which is exactly why convex formulations are prized - the conditions turn from a filter into a certificate. For a nonconvex problem, a KKT point may be a local minimum, a local maximum, or a saddle. It is a candidate, not an answer. ## Practical use Beyond theory, KKT is a diagnostic. Given a claimed solution and its multipliers you can check each condition arithmetically: is it feasible, are all inequality multipliers nonnegative, does each multiplier vanish where its constraint is slack, does stationarity balance. A solver reporting a negative multiplier on an inequality, or a positive multiplier on a constraint that is strictly slack, is reporting an inconsistency worth investigating before the result is trusted. The multipliers that come back nonzero also tell you which requirements are actually shaping the answer - typically a small fraction of the ones written down.