Every linear programme carries a second one whose variables are prices, and the simplex method has been solving both of them the whole time. That single fact is a tool: a final tableau will price a resource, bound the change that price survives, price a column that is not in the model yet, and reject a new constraint without a re-solve. The course ends where duality stops being bookkeeping — a Pareto front traced by one parameter, and a zero-sum game whose minimax theorem is strong duality with different nouns.
Overview
By the end you can write the dual of anything, prove a bound without solving, certify or refute a claimed optimum without solving, read four different sensitivity answers off one final tableau, re-optimise after a change, and recognise strong duality when it turns up wearing a different name.
One dual variable per constraint, one dual constraint per variable, the direction and sign rules applied correctly to a mixed-form problem — and each dual variable given units of objective per unit of right-hand side.
Weak duality from any feasible pair, with the two sign restrictions that produce it; and complementary slackness used to confirm or refute a claimed primal optimum without running the algorithm.
y = c_BᵀB⁻¹ from the columns that started as the identity, checked for dual feasibility column by column and for bᵀy = z* exactly.
A shadow price with its right-hand-side range, an objective coefficient's range for a basic and for a nonbasic variable, and the statement of which of the solution and its value changes in each case.
Price a proposed new activity by cₙₑw − yᵀaₙₑw and test a proposed new constraint at the current optimum — and re-optimise by the dual simplex only when one of them says you must.
Trace a two-objective Pareto front by one parameter with exact breakpoints, and solve a zero-sum game as a pair of dual linear programmes whose common value is the value of the game.
Lessons
Construction, then the two theorems, then the four things a final tableau will tell you, then the two ways to use them on a problem that has changed, and last two places where duality is the content rather than the method.
Linear programming models and the simplex method from this path, in particular the final tableau read as the inverse of the basis times the original data, plus inverse matrices, matrix arithmetic, gaussian elimination, piecewise functions, interval and set-builder notation and linear inequalities from algebra, and nothing at all from discrete mathematics, which is what makes this the last course an algebra-only reader can take and a complete story on its own.
Put units on every dual variable the moment you write it. A dual variable whose units you cannot state is a dual variable you have paired with the wrong constraint, and the check costs ten seconds.
Never quote a shadow price without its range. The lab prints them together and this course never separates them; a price outside its range is not approximately right, it is a different number.
Before running the dual simplex, try to say which basis the answer will end up in. You are usually wrong, and being wrong deliberately is how the ratio test on the objective row stops being a rule and starts being a reason.
Do “Complementary Slackness” with a degenerate instance as well as a clean one. It is where degeneracy first bites, and a reader who has only seen the conditions behave will assert the converse for years.
Farkas' lemma and the theorems of the alternative as a topic. Strong duality is obtained constructively from the final tableau, which is the version a reader can execute; the separating-hyperplane route is named and declined.
Lagrangian duality, and duality for anything non-linear. Both need machinery this path refuses.
Parametric analysis of the constraint matrix A. Ranging here is on b and on c; changing A changes the basis matrix itself and is a different subject.
Degeneracy's full effect on sensitivity. This course says what goes wrong — a tight constraint can have a zero price, and a range can end at the value the right-hand side already has — and does not develop the theory of the multiple dual solutions that cause it.
Interpreting a shadow price as a market price. The course says repeatedly what it is (a rate of change over a range, inside this model) and declines the economics.