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Calculus II: Vector Calculus and Multivariable Analysis / Multivariate Optimisation

Overview

Every portfolio construction problem, every model calibration, and every risk-budget allocation reduces to optimising a function of several variables — subject or not to constraints. This topic develops the complete toolkit: finding where a smooth objective function has no first-order gradient, classifying those critical points using the Hessian matrix (the multivariable analogue of the second derivative test), handling global extrema on closed bounded domains, and solving constrained problems via Lagrange multipliers. The last lesson supplies the proof that the Lagrange method is not a recipe but a geometric consequence of gradient perpendicularity.

By the end of this topic you should be able to:

  • locate critical points by solving ∇f=0\nabla f = \mathbf{0};
  • classify critical points as local max, local min, or saddle using the Hessian determinant test;
  • find absolute extrema on a compact domain via the boundary–critical-point method;
  • solve equality-constrained optimisation problems using the method of Lagrange multipliers;
  • interpret the Lagrange multiplier λ\lambda as a shadow price — the marginal value of relaxing the constraint by one unit;
  • prove the Lagrange condition ∇f=λ∇g\nabla f = \lambda\nabla g from first principles.

In this lesson

  1. Contents
  2. The big picture

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