Convex Optimization

A function $f$ is convex if $f(\lambda x + (1-\lambda) y) \le \lambda f(x) + (1-\lambda) f(y)$ for all $x, y$ and $\lambda \in [0, 1]$. A set is convex if it contains every line segment between its points.

Convex optimization minimizes a convex function over a convex set. The defining property: any local minimum is a global minimum. There are no spurious basins to get stuck in.

Convex problems are solvable by dependable polynomial-time algorithms (interior-point methods, gradient-based methods with global guarantees). Linear programs, quadratic programs, semidefinite programs, second-order cone programs are all convex.

Recognizing convexity matters: $\|x\|^2$ is convex; $\log x$ is concave (so $-\log x$ is convex); sums and pointwise maxima of convex functions are convex; affine compositions preserve convexity. In finance, mean-variance portfolio optimization is convex; option pricing via static replication is a linear program.