MATH2070 Chap.6 KKT and Constrained Optimisation
KKT and Constrained Optimisation
Constraints allow an objective gradient to be balanced by boundary normals rather than vanish. The Lagrangian creates that balance, while the KKT system adds primal feasibility, dual feasibility and complementary slackness. Each inequality is active with zero slack or inactive with zero multiplier, creating a systematic active-set analysis. Sign conventions must remain consistent.
Under suitable regularity and convexity, a feasible KKT point is globally optimal; without convexity it remains a candidate requiring further comparison. Equality multipliers are unrestricted, while inequality multiplier signs depend on the chosen constraint convention. An active-set table turns complementary slackness into testable cases and permits early rejection when feasibility, activity or sign assumptions fail.
Constraint qualifications explain when the first-order conditions are necessary by preventing degenerate descriptions of the local boundary. Multiplier interpretation connects constrained calculus back to shadow prices, but any sensitivity statement must preserve the convention used in the Lagrangian.
What this chapter covers
- 01
Lagrangians
- 02
Stationarity
- 03
Primal feasibility
- 04
Dual feasibility
- 05
Complementary slackness
- 06
Active sets
- 07
Convex sufficiency
Worked example · free
Project a point onto a half-plane
- 1State the inequality convention and Lagrangian.
- 1Solve stationarity.
- 1Test active and inactive cases.
- 1Use convexity for global optimality.
Key terms
- Lagrangian
- The objective plus constraint functions weighted by multipliers.
- Primal feasibility
- Satisfaction of the original equality and inequality constraints.
- Dual feasibility
- Satisfaction of multiplier sign conditions under the chosen convention.
- Complementary slackness
- The product condition forcing each inequality to be active or its multiplier to vanish.
- Active set
- The collection of inequality constraints satisfied at equality at a candidate. Hypothesising an active set converts complementary slackness into solvable cases that must still pass feasibility and multiplier-sign checks.
- Constraint qualification
- A regularity condition ensuring that the active constraints describe local feasible directions well enough for KKT necessity. It prevents degenerate formulations from invalidating the multiplier argument. The condition concerns whether boundary gradients provide a faithful local description, not whether the candidate is optimal.
KKT and Constrained Optimisation FAQ
What are the four KKT groups?
They are stationarity of the Lagrangian, primal feasibility of the original constraints, dual feasibility of inequality multipliers and complementary slackness between each multiplier and its constraint.
How should active-set cases be tested?
Assume each inequality active or inactive, solve stationarity with the corresponding zero slack or zero multiplier, then reject any case that violates its activity assumption, primal feasibility or multiplier sign.
When is a KKT point guaranteed to be global?
For a convex objective over a convex feasible set, under an appropriate constraint qualification, a KKT point is globally optimal. Strict convexity can add uniqueness; non-convex problems need further checks.
Exam move
Write the inequality convention before forming the Lagrangian. If constraints use h(x) less than or equal to zero in a minimisation problem, record the associated multiplier sign and preserve it throughout. List stationarity, primal feasibility, dual feasibility and complementary slackness as four separate groups before solving any equation. Keep equality multipliers apart because they are unrestricted in sign.
For a small number of inequalities, make an active-set table. Each row states which constraints bind, which multipliers are zero and which equations remain. Solve the easiest implications first, then reject a row immediately when it violates its assumed activity, an original constraint or a multiplier sign. This is more dependable than guessing that the visually important boundary must be active.
Verify every complementary product even when the candidate appears geometrically plausible. Use geometry to support, not replace, the equations. Sketch level curves and feasible boundaries, mark active normals and predict the gradient balance. For distance problems, locate the unconstrained minimiser first; if it is feasible, inactive multipliers should emerge, while an infeasible target suggests projection to a boundary.
Add examples with several candidate active sets so that case selection becomes deliberate rather than automatic. Finish every solution by identifying the scope of the conclusion. With a convex objective, convex inequality functions, affine equalities and a valid qualification, a feasible KKT point can provide a global optimum; strict convexity may give uniqueness.
Without those conditions, record it as a candidate and compare objective values or use further local analysis. Keep an error ledger for convention, stationarity, feasibility, complementarity and sufficiency. Revisit incorrect examples by changing only the constraint direction, because observing how the Lagrangian and multiplier sign change is a strong test of genuine understanding.
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