Convex Optimization
A convex optimization problem minimizes a convex function over a convex set — local minima are global, and the dual/KKT machinery is reliable. Most honest portfolio problems try to stay here.
Definition
f convex, constraints convex ⇒ any local min is global. QP mean-variance with linear constraints is convex. Cardinality (‘pick 10 names’) is not. Machine-learning losses (logistic, hinge, many regularizers) are built to stay convex.
Why it matters
If your optimizer is non-convex, the number it returns is a local story. Risk systems that assume uniqueness will lie.
Case
Markowitz QP vs a ‘maximize Sharpe with a 5-name constraint.’ The second needs heuristics; do not treat the output as the unique optimum.
How to read it
Ask whether the objective and feasible set are convex. If the vendor says ‘AI optimizer’ and cannot answer, assume local minima.
Key papers
Landmark references with real publisher or arXiv links — not token-overlap noise.
- Convex OptimizationBoyd, Vandenberghe · 2004 · Cambridge University PressThe standard reference (free PDF).