Skip to content
Linear Algebra, Optimisation and Numerical Methods / Optimisation

Unconstrained Optimisation

Quadratic utility w⊤μ−12λw⊤Σw\mathbf{w}^\top\boldsymbol\mu-\frac12\lambda\mathbf{w}^\top\Sigma\mathbf{w} and squared pricing error ∥p(θ)−pmkt∥2\lVert\mathbf{p}(\boldsymbol\theta)-\mathbf{p}_{\mathrm{mkt}}\rVert^2 are unconstrained optimisation problems once the decision variable is free. Critical points, the Hessian second-order test, and a carefully stated descent method (gradient or Newton) turn those objectives into something you can solve. This note synthesises the calculus toolkit into a finance-facing unconstrained engine; constraints arrive in the next lesson.
Learning objectives. By the end of this note you should be able to:
  • find critical points by solving ∇f=0\nabla f=\mathbf{0};
  • classify them via positive / negative / indefinite Hessians;
  • state gradient descent and Newton's method for unconstrained minimisation;
  • maximise a quadratic utility or minimise a squared pricing error on a small example.

In this lesson

  1. Critical points and the gradient
  2. The Hessian test
  3. Descent methods

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or browse the Linear Algebra, Optimisation and Numerical Methods syllabus.