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cla

Concept Overview

Markowitz’s Critical Line Algorithm in the Bailey-Lopez de Prado formulation: the exact solution to the constrained mean-variance problem with inequality bounds on every weight. Rather than calling a general quadratic solver it walks the efficient frontier from the maximum-return corner, computing each turning point where an asset enters or leaves the free set. That yields the whole frontier rather than one point on it, and it terminates — which quadratic solvers on near-singular covariance matrices frequently do not.

When to Use

Use it when you need the full efficient frontier, when weight bounds are binding, or when a general optimiser is returning unstable or non-converging weights on an ill-conditioned covariance. If you only want one portfolio and the covariance is well behaved, portfolio_optimization is the shorter path. If the covariance itself is the problem, prefer hrp, which never inverts it. CLA still needs expected returns, so it inherits their estimation error.

Mathematical Foundations

MVO Objective

$\min_w\;\frac{1}{2}w^T\Sigma w-\lambda\mu^T w$

Budget Constraint

$\mathbf{1}^T w=1$

Usage Examples

Rust

Prepare covariance for CLA

use nalgebra::DMatrix;
use openquant::cla::covariance;

let returns = DMatrix::from_row_slice(3, 2, &[0.01, 0.02, -0.01, 0.01, 0.015, 0.03]);
let sigma = covariance(&returns);

API Reference

Python API

cla.allocate_cla

Rust API

CLA
covariance
ReturnsEstimation

Risk Notes and Caveats

CLA behavior depends on weight bounds and return estimates.
Use robust covariance estimators when sample size is small.

cla