fast-minimum-variance 0.3.0

Constructing minimum variance portfolios

fast-minimum-variance: Solving Minimum Variance Portfolios Fast

Overview

fast-minimum-variance is a Python library for computing long-only minimum variance portfolios without ever forming the sample covariance matrix. By operating directly on the returns matrix $R \in \mathbb{R}^{T \times N}$, it exposes a clean hierarchy of solvers — from an exact direct KKT solve to matrix-free Krylov methods — that scale gracefully as $N$ grows.

The core insight is that minimising portfolio variance is equivalent to minimising $|Rw|^2$, which can be evaluated using two matrix-vector products $w \mapsto R^\top(Rw)$ without constructing $R^\top R$ explicitly. This reframing connects the portfolio optimisation literature directly to Krylov subspace methods.

The long-only constraint $w \geq 0$ is handled throughout via an active-set method: solve the unconstrained problem on the current active set, drop assets with negative weights, and repeat. The process terminates in at most $N$ iterations.

Solvers

Solver	Module	Method	Notes
solve_kkt	kkt	Direct KKT via numpy.linalg.solve	Exact; baseline for accuracy comparisons
solve_minres	krylov	MINRES on the indefinite KKT system	Matrix-free capable; handles indefiniteness correctly
solve_cg	krylov	CG in the constraint-reduced space	Positive-definite reduced system; no indefinite solver needed
solve_cvxpy	cvx	General-purpose convex solver via CVXPY	Reference implementation; slowest but most flexible

All solvers return a weight vector $w \in \mathbb{R}^N$ satisfying $\sum_i w_i = 1$ and $w_i \geq 0$.

Quick Start

from fast_minimum_variance.random import make_returns
from fast_minimum_variance.kkt import solve_kkt
from fast_minimum_variance.krylov import solve_cg, solve_minres
from fast_minimum_variance.cvx import solve_cvxpy

# Generate a synthetic return matrix: 500 daily returns, 20 assets
R = make_returns(T=500, N=20, seed=42)

# Solve with any of the available solvers
w_kkt = solve_kkt(R)  # exact KKT solve
w_minres = solve_minres(R)  # MINRES on the indefinite KKT system
w_cg = solve_cg(R)  # CG in the constraint-reduced space
w_cvxpy = solve_cvxpy(R)  # CVXPY reference

# All solutions satisfy the portfolio constraints
assert abs(w_kkt.sum() - 1.0) < 1e-8
assert (w_kkt >= 0).all()

The KKT System

The equality-constrained minimum variance problem yields the $(N+1) \times (N+1)$ KKT system:

$$\begin{pmatrix} 2R^\top R & \mathbf{1} \cr \mathbf{1}^\top & 0 \end{pmatrix} \begin{pmatrix} w \cr \lambda \end{pmatrix} = \begin{pmatrix} \mathbf{0} \cr 1 \end{pmatrix}$$

This system is symmetric but indefinite — the zero in the bottom-right corner of the KKT matrix introduces a negative eigenvalue. This rules out standard CG on the full system, but it opens the door to MINRES. Alternatively, the CG solver eliminates the constraint entirely by parameterising $w = w_0 + Pv$ where $P$ spans the null space of $\mathbf{1}^\top$, yielding a positive-definite reduced system of size $(N-1) \times (N-1)$.

Installation

pip install fast-minimum-variance

For development:

git clone https://github.com/Jebel-Quant/fast_minimum_variance
cd fast_minimum_variance
make install

Requirements

• Python 3.11+
• numpy
• scipy
• cvxpy

Citing

If you use this library in academic work or research, please cite:

@software{fast_minimum_variance,
  author  = {Schmelzer, Thomas},
  title   = {fast-minimum-variance: Solving Minimum Variance Portfolios Fast},
  url     = {https://github.com/Jebel-Quant/fast_minimum_variance},
  year    = {2026},
  license = {MIT}
}

License

MIT License — see LICENSE for details.