Shadow price

A shadow price is the change in an optimization problem’s best achievable objective value from relaxing a constraint by one unit, under specified conditions.

Lagrange multiplier: A scalar attached to a constraint in a constrained optimization problem, used to characterize how the optimum responds to changes in that constraint. In smooth problems, the multiplier often equals the constraint’s shadow price, up to sign convention.
Mathematical optimization: The study of selecting the best feasible decision according to a specified objective and constraints. Shadow prices are defined relative to an optimization problem’s objective and feasible set.
Economic scarcity: The condition in which limited resources cannot satisfy all competing uses or wants. A resource constraint has a positive shadow price when relaxing it can improve the objective.
Market price: The monetary amount at which a good or service is bought and sold in a market. A shadow price can exist without a traded market price and need not equal one.
Nonlinear programming: Optimization involving objective functions or constraints that are nonlinear in the decision variables. Its local multipliers may fail to describe finite relaxations when curvature or nonconvexity matters.
Duality (optimization): A relationship between an optimization problem and a second problem whose solutions provide bounds or characterize the original optimum. Dual variables give shadow prices a precise interpretation as values assigned to constraints.
Constraint (mathematics): A condition restricting which values or decisions are permitted in a mathematical problem. The constraint is what gets relaxed, and its units determine the price’s units.
Resource allocation: The assignment of limited resources among competing activities or uses. Optimization models use shadow prices to compare the marginal value of added resources.
Accounting cost: A cost recorded using financial accounting rules, based on recognized expenditures and allocations. Accounting costs record transactions; shadow prices measure modeled marginal effects.
Nonconvex optimization: Optimization problems whose feasible sets or objective structure do not satisfy convexity conditions. Multiple local optima can make a single shadow-price interpretation unreliable.

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