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Calculus II: Vector Calculus and Multivariable Analysis / Multivariate Optimisation

Constrained Optimisation and Lagrange Multipliers

The mean-variance efficient frontier is not an unconstrained critical point — it is the solution of a family of problems of the form "maximise expected return subject to variance equal to σ02\sigma_0^2". The constraint locks the weights onto a curve (or surface), and the unconstrained gradient ∇μ\nabla\mu is no longer the operative object. The method of Lagrange multipliers converts a constrained problem into an extended system of equations whose solutions are the constrained critical points. The Lagrange multiplier λ\lambda has a precise economic meaning: it is the shadow price of the constraint, measuring how the optimal objective value changes when the constraint is relaxed by one unit. This note develops the method for one and two constraints, works through the portfolio application, and establishes the shadow price interpretation.
Learning objectives. By the end of this note you should be able to:
  • state the Lagrange condition ∇f=λ∇g\nabla f = \lambda\nabla g and form the Lagrangian L=f−λg\mathcal{L} = f - \lambda g;
  • solve a one-constraint optimisation problem using the Lagrange system;
  • extend the method to two constraints and three variables;
  • interpret λ\lambda as the shadow price: the rate at which f∗f^* changes as the constraint level cc is relaxed;
  • apply Lagrange multipliers to the mean-variance frontier.

In this lesson

    1. The Lagrange Condition
    1. The Method — One Constraint in Two Variables
    1. Two Constraints — Three Variables
    1. The Shadow Price Interpretation
    1. Practical Example — Box of Maximum Volume

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