Shadow prices and sensitivity — the same idea, twice
A Lagrange multiplier's marginal value and a linear program's dual variable are the same concept — met first in Chapter 6, formalized again in Chapter 8.
Chapter 6 introduces the idea in general nonlinear terms — a Lagrange multiplier measures how much the optimal objective value would change per unit of relaxing its constraint. Chapter 8 formalizes exactly the same idea for linear programs specifically, where it comes with an exact usable range (how far a cost or resource can move before the optimal basis itself changes) and shows up automatically in any solver's sensitivity report.
First meeting — the Lagrange multiplier's meaning
Term
Lagrangian
The objective combined with every constraint, each weighted by its own multiplier.
Formalized for linear programs
How much the optimal objective value changes per unit change in a constraint's right-hand side.
The interval a single cost or resource coefficient can move through while the current optimal vertex, and its shadow prices, stay valid.
How far a cost or resource coefficient can move before the optimal basis itself would change.
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