# Notes - Machine Learning MT23, Convex optimisation

> Source: https://ollybritton.com/notes/uni/part-a/mt23/machine-learning/notes/convex-optimisation/ · Updated: 2026-06-08 · Tags: uni, notes

- [Course - Machine Learning MT23](https://ollybritton.com/notes/uni/part-a/mt23/machine-learning/)
	- [Notes - Continuous Mathematics HT23, Convexity](https://ollybritton.com/notes/uni/prelims/ht23/continuous-mathematics/notes/convexity/)

### Flashcards
What does it mean for a set $C \subseteq \mathbb R^D$ to be convex?::

For all $x, y \in C$ and $\lambda \in [0, 1]$,
$$
\lambda \cdot \pmb x + (1 - \lambda) \cdot \pmb y \in C
$$

In terms of a matrix $\pmb A \in \mathbb R^{m \times n}$ and $\pmb b \in \mathbb R^m$, what is a polyhedron?::

The set of points
$$
P = \{ x\in \mathbb R^n \mid \pmb A \cdot \pmb x \le \pmb b\}
$$

What does it mean for a function $f : \mathcal C \to \mathbb R$ to be convex?::

For all $\pmb x, \pmb y \in \mathcal C$ where $f$ defined and $0 \le \lambda \le 1$,
$$
f(\lambda\pmb x + (1 - \lambda)\pmb y) \le \lambda f(\pmb x) + (1-\lambda)f(\pmb y)
$$

What makes convex optimisation problems nice?::

All locally optimal points are globally optimal.

What is a subderivative of a convex function at some non-differentiable point $x_0$?::

A real number $c$ such that, for all $x$,
$$
f(x) \ge f(x_0) + c(x-x_0)
$$

Suppose we have a convex function $f : \mathbb R^n \to \mathbb R$. @Define a subgradient $\pmb g$ of $f$ at point $\pmb w_0$.::

Any $\pmb g$ satisfying
$$
f(\pmb w) \ge f(\pmb w_0) + \pmb g^\top (\pmb w - \pmb w_0)
$$

What are subgradient methods in the context of gradient descent?::

Convex optimisation methods that generalise gradient descent to non-differentiable functions via subderivatives.

What is the idea behind projected gradient descent for approximating solving convex optimisation problems?::

Use standard gradient descent approaches but use a "projection operator" to ensure that it doesn't leave the feasible region.

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