a function for which, given any two points, the function between those points sits at (lines are convex!) or below the plane given those points

constituents

For \(f: \mathbb{R}^{n} \to \mathbb{R}\)

requirements

\begin{equation} f\qty(\theta x + \qty(1-\theta) y) \leq \theta f\qty(x) + \qty(1-\theta) f\qty(y) \end{equation}

strictly convex is the strict inequality

additional information

some convex functions

• affine: \(ax + b\)
• exponential: \(e^{ax}\)
• powers: \(x^{\alpha}\) for \(\alpha \geq 1\) or \(\alpha \leq 0\)
• \(|x|^{p}\) for \(p > 1\)
• relu
• any norm
• sum of squares: \(\qty {x}^{2}_{2} = x_1^{2} + … + x_{n}^{2}\)
• max function: \(\max \qty(x) = \max \qty {x_1 \dots x_{n}}\)
• softmax: \(\log \qty(\exp x_1 + \dots + \exp x_{n})\)
• general affine function: \(f\qty(X) = tr\qty(A^{T} X) + b\) (“an inner product”)
• spectral norm: \(f\qty(X) = \norm{X}_{2} = \sigma_{\max}\qty(X)\) (the maximum singular value of \(X\)

some concave fuctions

• affine
• powers
• logs: \(\log x\)
• entropy: \(- x \log x\)
• negative part (opposite relu)
• log determinant: \(f\qty(X) = \log \text{det} X\)

EE364A.stanford.edu

Lecture

• SU-EE364a JAN062026
• SU-EE364A JAN082026

Suppose \(f\) is convex on \(\mathbb{R}^{n}\), with domain \(\text{dom } f\). We can make an extended-value extension:

\begin{equation} \tilde{f} \qty(x) = \begin{cases} f\qty(x), x \in \text{dom }f \\ \infty, \text{ otherwise } \end{cases} \end{equation}

This could simplify notation .

• SU-STS115 JAN052025
• SU-STS115 JAN072025

requirements

A perspective function is:

\begin{equation} P\qty(x,t) = \frac{x}{t} \end{equation}

\(P: \mathbb{R}^{n+1} \to \mathbb{R}^{n}\).