Convex Optimization

Motivation: why this matters in quant finance

Portfolio construction, calibration, and regularised estimation all become safer when the objective is convex. Convexity turns local improvement into global reliability.

The informal idea

A convex optimisation problem minimises a bowl-shaped objective over a bowl-shaped feasible set. Lines between feasible points stay feasible, and the objective along those lines never arches above the chord.

Formal definitions

A set $C$ is convex if $\theta x+(1-\theta)y\in C$ for $x,y\in C$ and $0\le\theta\le1$. A function is convex if $f(\theta x+(1-\theta)y)\le\theta f(x)+(1-\theta)f(y)$.

Key properties

Local minima are global

This is the central computational value of convexity.

First-order conditions certify optima

For differentiable unconstrained convex $f$, $\nabla f(x^*)=0$ is sufficient.

Quadratic risk problems are convex when covariance is positive semidefinite

That is why mean-variance optimisation is tractable.

Worked example

Minimising $f(w)=\frac{1}{2}\sigma^2w^2-\mu w$ is convex when $\sigma^2>0$. The first-order condition gives $w^*=\mu/\sigma^2$.

Common confusions and pitfalls

"Every optimisation problem solved by a computer is convex." Many calibration and trading problems are non-convex.

"Convex means linear." Quadratic, norm, and exponential losses can be convex too.

"Constraints are bookkeeping." Constraints define the feasible set and can change the solution.

Where this goes next

• Lagrange Multipliers and KKT Conditions: gives optimality conditions for constrained convex problems.
• Quadratic Programming: specialises convex optimisation.
• Mean-Variance Optimisation: is the portfolio application.