Lagrange multiplier

A variable introduced to enforce a constraint in an optimization problem. It turns constrained extrema into stationary points of a combined function.

Lagrangian: A function formed by combining an objective with constraints multiplied by auxiliary variables. Its stationary-point equations encode both the objective’s gradient and the constraint.
Constrained optimization: Optimization in which candidate solutions must satisfy specified restrictions. Lagrange multipliers are a standard method within this broader problem class.
Shadow price: The marginal change in an optimal objective value when a constraint’s bound changes. An optimal multiplier often measures the value of relaxing its associated resource constraint.
Penalty method: An optimization technique that adds a cost for violating constraints to the objective function. Unlike exact multiplier equations, penalties allow constraint violations during intermediate steps.
Gradient: A vector of partial derivatives pointing in the direction of greatest local increase of a scalar function. At a constrained extremum, the objective gradient aligns with the constraint gradient.
Equality constraint: A restriction requiring a function of the variables to equal a specified value. The classical method handles smooth equality constraints.
Economic equilibrium: A state in which economic choices and market interactions satisfy specified balance conditions. Utility-maximizing agents use multiplier conditions to derive marginal trade-offs.
Projection method: An algorithm that repeatedly maps candidate points onto a constraint set or a related feasible region. It enforces feasibility geometrically rather than through an auxiliary multiplier.
Stationary point: A point where a differentiable function’s derivative or gradient is zero. The method searches for stationary points of the combined function.
Feasible set: The set of points satisfying every constraint in an optimization problem. The desired extremum must lie within this restricted set.

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