📈 cvxcla - Critical Line Algorithm for Portfolio Optimization

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📋 Overview

cvxcla is a Python package that implements the Critical Line Algorithm (CLA) for portfolio optimization. The CLA efficiently computes the entire efficient frontier for portfolio optimization problems with linear constraints and bounds on the weights.

The same parametric active-set engine also traces the LASSO regularisation path (the Lasso class) — including general inequality constraints Gβ ≤ h and the non-negative LASSO β ≥ 0 — so the Critical Line Algorithm and the LARS/LASSO homotopy come out as two instances of one path-following engine.

The Critical Line Algorithm was introduced by Harry Markowitz in The Optimization of Quadratic Functions Subject to Linear Constraints and further described in his book Portfolio Selection.

The algorithm is based on the observation that the optimal weights are a piecewise linear function of the return tilt λ. The CLA computes the turning points (corners) where that path bends, and the whole frontier is recovered from them exactly. In the (variance, return) plane the frontier is a chain of parabolic arcs that meet with continuous slope, which is why its corners are invisible on a frontier plot but show up as kinks on a LASSO coefficient path.

I gave the plenary talk at EQD's Singapore conference.

🧮 Why the Algorithm Works

The Markowitz problem is a quadratic program parametrized by a return tilt λ (a weight on expected return, not a return target or a risk aversion):

min  ½ wᵀΣw - λ · μᵀw
s.t. Aw = b,  Gw ≤ h,  lb ≤ w ≤ ub

where Aw = b are linear equalities (the budget, sector/factor neutrality, …) and Gw ≤ h are linear inequalities (group or sector exposure caps; a ≥ floor is the negated row). As λ sweeps from ∞ (maximize return) down to 0 (minimize variance), the solution traces the entire efficient frontier. The key insight is that between consecutive events, the optimal weights are a linear function of λ:

w(λ) = α + λ · β

This holds because the KKT optimality conditions are linear in λ whenever the active set — which assets sit at their bounds — is fixed. The algorithm exploits this in three steps:

1. Start at λ = ∞, where the portfolio concentrates on the highest-return asset within bounds (init_algo).

2. Solve the KKT system for the current active set to find α and β by block elimination, then decrease λ until one of two events occurs (main loop in cla.py):

   • a free asset hits its bound (leaves the free set), or
   • a blocked asset's KKT multiplier changes sign (enters the free set).

   General inequality rows Gw ≤ h add the row analogue of these two events: an inactive row becomes binding when its slack reaches zero, and an active row is released when its multiplier reaches zero. An active row is carried through the same block-eliminated solve as an extra equality row, so the structure is preserved.

3. Update the active set (exactly one asset or constraint row changes status) and repeat until λ ≤ 0.

Because only one coordinate changes per step and each step requires only a single linear solve, the algorithm traces the full frontier cheaply and exactly — no approximation needed. Ties are broken by a Bland-style lowest-index rule, which keeps the walk deterministic and finite.

The four events are the same ones the LASSO homotopy uses, under different names (Schmelzer and Hastie, 2026, Table 2):

Event	Critical Line Algorithm	LASSO / LARS
P₁	an asset reaches a bound and leaves the free set	a coefficient crosses zero (the "leave" move)
P₂	a group or exposure row Gw ≤ h becomes tight	a linear inequality on β becomes tight
D₁	a blocked asset's reduced cost changes sign and it re-enters	an inactive correlation reaches λ (the "enter" move)
D₂	an active row's multiplier hits zero and it releases	an active inequality's multiplier hits zero

The only difference at the level of the algorithm is D₁: for a box the threshold is fixed at zero, while for the ℓ₁ penalty it moves with λ.

✨ Features

• Efficient computation of the entire efficient frontier
• Fluent builder API — CLA.problem(mean, cov).long_only().budget().trace() — as a readable alternative to the explicit constructor
• Box bounds on every weight, plus general linear equality constraints Aw = b (budget, dollar-neutral, sector/factor neutrality) and general linear inequality constraints Gw ≤ h (group or sector exposure caps)
• Leverage (gross-exposure) caps ‖w‖₁ ≤ c, traced exactly (130/30 books, capped dollar-neutral books)
• Factor covariance backend: exact frontiers for diagonal-plus-low-rank covariances (factor models, RMT-cleaned matrices) in O(nk) memory via the Woodbury identity
• Visualization of the efficient frontier using Plotly
• Computation of the maximum Sharpe ratio portfolio
• LASSO regularisation path through the same engine (Lasso), with the same fluent builder, general inequality constraints Gβ ≤ h, per-coefficient bounds (caps, and signs on any of the coefficients), and equality rows Aβ = b
• Fully tested and documented codebase

🚀 Installation

Using pip

pip install cvxcla

To include plotting support (Plotly and Kaleido):

pip install cvxcla[plot]

Development Setup

To set up a development environment:

1. Clone the repository:

   git clone https://github.com/cvxgrp/cvxcla.git
   cd cvxcla
   

2. Create a virtual environment and install dependencies:

   make install
   

This will:

• Install the uv package manager
• Create a Python 3.12 virtual environment
• Install all dependencies from pyproject.toml

🔧 Usage

Here's a simple example of how to use cvxcla to compute the efficient frontier:

import numpy as np

# Set a seed for reproducibility
np.random.seed(42)
from cvxcla import CLA

# Define your portfolio problem
n = 10  # Number of assets
mean = np.random.randn(n)  # Expected returns
cov = np.random.randn(n, n)
covariance = cov @ cov.T  # Covariance matrix
lower_bounds = np.zeros(n)  # No short selling
upper_bounds = np.ones(n)  # No leverage

# Create a CLA instance
cla = CLA(
    mean=mean,
    covariance=covariance,
    lower_bounds=lower_bounds,
    upper_bounds=upper_bounds,
    a=np.ones((1, n)),  # Fully invested constraint
    b=np.ones(1),
)

# Access the efficient frontier
frontier = cla.frontier

# Get the maximum Sharpe ratio portfolio
max_sharpe_ratio, max_sharpe_weights = frontier.max_sharpe
print(f"Maximum Sharpe ratio: {max_sharpe_ratio:.6f}")
# Print first few weights to avoid long output
print(f"First 3 weights: {max_sharpe_weights[:3]}")

Maximum Sharpe ratio: 2.946979
First 3 weights: [0.         0.         0.08509841]

The same problem reads more declaratively through the fluent builder, reached via CLA.problem(...). Each step maps onto one constructor argument, and the terminal .trace() returns the identical CLA:

# Same fully-invested, long-only problem as above, assembled fluently.
built = CLA.problem(mean, covariance).long_only().budget().trace()

# Pure sugar over the constructor: it returns the identical frontier.
assert built.frontier.max_sharpe[0] == cla.frontier.max_sharpe[0]

.bounds, .equality, and .inequality add general bounds and constraints, as in the examples below.

For visualization, you can plot the efficient frontier:

# Plot the efficient frontier
fig = frontier.plot(volatility=True)
fig.show()

Group and sector constraints

Pass g and h to add general inequality rows Gw ≤ h on top of the box bounds and the budget. For example, to cap the combined weight of the first three assets (a sector) at 40%:

import numpy as np
from cvxcla import CLA

n = 10
rng = np.random.default_rng(42)
mean = rng.standard_normal(n)
factor = rng.standard_normal((n, n))
covariance = factor @ factor.T  # symmetric positive definite

# sum of the first three weights ≤ 0.40
g = np.zeros((1, n))
g[0, :3] = 1.0
h = np.array([0.40])

cla = CLA(
    mean=mean,
    covariance=covariance,
    lower_bounds=np.zeros(n),
    upper_bounds=np.full(n, 0.35),
    a=np.ones((1, n)),  # fully invested
    b=np.ones(1),
    g=g,  # inequality matrix  G  (p x n)
    h=h,  # inequality vector  h  (p,)
)

# every turning point now satisfies the sector cap
assert all(tp.weights[:3].sum() <= 0.40 + cla.tol for tp in cla.turning_points)

g/h default to None, so omitting them recovers the equality-only problem unchanged. Stack multiple rows in g for several caps at once, and express a ≥ floor by negating a row (-Gw ≤ -h).

Leverage (gross-exposure) caps

Pass leverage=c (or chain .leverage(c) on the builder) to cap the gross exposure ‖w‖₁ = Σ|wᵢ| ≤ c. With a fully-invested budget, c = 1.3 is a 130/30 book:

import numpy as np
from cvxcla import CLA

n = 10
rng = np.random.default_rng(42)
mean = rng.standard_normal(n)
factor = rng.standard_normal((n, n))
covariance = factor @ factor.T  # symmetric positive definite

cla = (
    CLA.problem(mean, covariance)
    .bounds(-0.2, 0.45)  # shorts allowed down to -20% per asset
    .budget()  # fully invested
    .leverage(1.3)  # at most 130% long + short
    .trace()
)

assert all(np.abs(tp.weights).sum() <= 1.3 + cla.tol for tp in cla.turning_points)

The 1-norm is polyhedral, so the frontier is still traced exactly. Internally each asset whose box straddles zero is split into a long and a short leg, wᵢ = uᵢ − vᵢ, and the cap becomes one extra inequality row Σuᵢ + Σvᵢ ≤ c. The turning points are reported in the original weights. Assets that can only be long (or only short) are not split, so the cap is redundant for a long-only, fully-invested book.

Factor models at scale

For diagonal-plus-low-rank covariances (factor risk models, eigenvalue-clipped covariances) pass a FactorCovariance instead of a dense matrix; the algorithm then never forms an n-by-n matrix:

import numpy as np
from cvxcla import CLA, FactorCovariance

rng = np.random.default_rng(42)
n, k = 10_000, 50
covariance = FactorCovariance(
    d=rng.uniform(0.1, 0.5, n),  # idiosyncratic variances
    u=rng.standard_normal((n, k)),  # factor loadings
    delta=rng.uniform(0.5, 2.0, k),  # factor variances, (k,) or (k, k)
)

See the factor backend documentation for the protocol, the math, and benchmarks against the dense path.

The LASSO through the same engine

The same path-tracer drives the LASSO regularisation path. Lasso plays the Gram matrix XᵀX and the vector Xᵀy the way the CLA plays the covariance and the mean, and traces the whole path from λ_max (where β = 0) down to the least-squares fit:

import numpy as np
from cvxcla import Lasso

rng = np.random.default_rng(0)
X = rng.standard_normal((60, 12))
y = X @ rng.standard_normal(12) + 0.1 * rng.standard_normal(60)

# plain LASSO path; .solution(lam) evaluates beta at any penalty
lasso = Lasso(x=X, y=y)
beta = lasso.solution(0.5 * lasso.lam_max)

# the same fluent builder as the CLA
lasso = Lasso.problem(X, y).trace()

It carries the same constraints the CLA does. Add general inequality rows Gβ ≤ h (with h > 0, e.g. per-group exposure caps), or restrict to the non-negative LASSO β ≥ 0 — where the ℓ₁ penalty collapses to the linear term λ·1ᵀβ, making it exactly the CLA's box-bounded parametric QP:

# group-exposure caps G beta <= h (cap features 0–3 and 4–7)
G = np.zeros((2, 12))
G[0, :4] = 1.0
G[1, 4:8] = 1.0
h = np.array([1.0, 1.0])
lasso = Lasso.problem(X, y).inequality(G, h).trace()

# non-negative LASSO (beta >= 0)
lasso = Lasso.problem(X, y).non_negative().trace()

Both return the exact path, validated breakpoint-by-breakpoint against a per-λ QP solver.

Homogeneous equality constraints Aβ = 0 are supported too, for example sum-to-zero coefficients or contrasts:

# sum-to-zero LASSO (1ᵀβ = 0); rows accumulate
lasso = Lasso.problem(X, y).equality(np.ones(12)).trace()
assert abs(lasso.solution(0.5 * lasso.lam_max).sum()) < 1e-10

The LASSO homotopy starts from β = 0, where an equality row is already active, so it cannot seed this path one coordinate at a time. The path is traced instead as one leverage-capped CLA on Σ = XᵀX, μ = Xᵀy and rescaled, β = w / λ, using the identity described next. Signs on any of the coefficients come along, since a bound of zero survives the rescaling.

Per-coefficient bounds and equality rows with a nonzero right-hand side go through the CLA in the penalty form: each coefficient is split into a long and a short leg, β = Sz with z ≥ 0, and the penalty λ‖β‖₁ becomes a cost of λ per unit of gross position, so the CLA's tilt is λ itself (covariance SᵀXᵀXS, expected returns −1, benchmark q = SᵀXᵀy). Caps, signs, Aβ = b and inequality rows Gβ ≤ h (with h ≥ 0) combine:

# beta_0..2 >= 0, and every |beta_j| <= 2
lower = np.full(12, -2.0)
lower[:3] = 0.0
upper = np.full(12, 2.0)
lasso = Lasso.problem(X, y).bounds(lower, upper).trace()

# the same, with a fit that must pass through the first observation
lasso = Lasso.problem(X, y).equality(X[0], y[0]).bounds(lower, upper).trace()
assert abs(X[0] @ lasso.solution(0.5 * lasso.lam_max) - y[0]) < 1e-9

# and with the net effect of features 0-3 capped, |beta_0 + ... + beta_3| <= 1
G = np.zeros((2, 12))
G[0, :4], G[1, :4] = 1.0, -1.0
lasso = Lasso.problem(X, y).equality(X[0], y[0]).inequality(G, np.ones(2)).bounds(lower, upper).trace()
assert abs(G[0] @ lasso.path[-1].beta) <= 1.0 + 1e-9

# a monotone fit, beta_0 <= beta_1 <= ... <= beta_11: homogeneous rows, h = 0
D = np.eye(11, 12) - np.eye(11, 12, k=1)
lasso = Lasso.problem(X, y).inequality(D, np.zeros(11)).trace()
assert all((D @ bp.beta <= 1e-12).all() for bp in lasso.path)

Homogeneous rows (h = 0) are all tight at β = 0, so they start on a degenerate vertex. They are traced twice, relaxed to h = ε and h = 2ε, and on a fixed sequence of active sets every knot is affine in h, so 2·path(ε) − path(2ε) is the exact path at h = 0. The result is returned only after it passes an optimality certificate: a linear program finds the best multipliers at every knot and segment midpoint, and the conditions must hold to rounding.

The two routes have these limits:

• Aβ = 0 takes signs (bounds of 0 or ±∞) but neither a finite cap nor inequality rows: the rescaling cannot carry them, and the penalty form starts on a degenerate vertex
• homogeneous inequality rows are declined if the two relaxed traces take different sequences of active sets, or if the extrapolated path fails its certificate
• Aβ = b with b ≠ 0 needs a unique smallest-ℓ₁ point on Aβ = b to start from; the fully invested 1ᵀβ = 1 has a whole face of them and is refused
• under Aβ = 0 the Gram XᵀX must be positive definite, so the constrained least-squares end is unique

Block-structured homogeneous rows, such as two groups that each sum to zero, make the first vertex degenerate and are refused.

One curve, two literatures

The link between the two classes is more than a shared engine. Under Σ = XᵀX and μ = Xᵀy, the gross-exposure-capped Markowitz program and the constrained LASSO trace the same piecewise-linear curve, under arbitrary linear equality and inequality constraints (Schmelzer and Hastie, 2026):

M_c:  min ½ wᵀΣw − μᵀw        s.t. ‖w‖₁ ≤ c,  Aw = b,  Gw ≤ h
L_λ:  min ½‖y − Xβ‖² + λ‖β‖₁  s.t.          Aβ = b,  Gβ ≤ h

• Theorem 1. Given general position and a constraint qualification, β(λ) solves M_c at c(λ) = ‖β(λ)‖₁. Wherever c is strictly decreasing, the two paths share the same breakpoints, visited in the same order.
• Budget versus tilt (Corollary 2). With homogeneous constraints (b = 0, h = 0), sweeping the tilt λ at a fixed leverage cap c (which is what CLA(leverage=c) does) gives the budget-indexed path rescaled: w_c(λ) = λ · w^M(c/λ).
• The frontier is a LASSO path (Proposition 3). The long-only, fully invested frontier is the non-negative LASSO path, rescaled radially by t = 1/λ.
• Degrees of freedom (Proposition 4). A turning point's fit has df = E[|F| − rank M_F], where F is the free set and M_F stacks the active constraint rows on it. For a long-only, fully invested frontier this is the expected number of holdings away from a bound, less one.

In cvxcla terms, a CLA with mean = Xᵀy, covariance = XᵀX and leverage = c reproduces the rescaled LASSO path, and so does a Lasso fitted on (X, y):

import numpy as np
from cvxcla import CLA, Lasso

rng = np.random.default_rng(0)
X = rng.standard_normal((40, 8))
y = X @ rng.standard_normal(8) + 0.3 * rng.standard_normal(40)

lasso = Lasso(x=X, y=y)
c = 0.8 * np.abs(lasso.path[-1].beta).sum()  # a binding leverage cap
cla = CLA(
    mean=X.T @ y,
    covariance=X.T @ X,
    lower_bounds=np.full(8, -50.0),
    upper_bounds=np.full(8, 50.0),
    a=np.zeros((0, 8)),
    b=np.zeros(0),
    leverage=c,
)
# every turning point w at tilt lam is lam * beta, where ||beta||_1 = c / lam

The identity carries statements about the curve, such as the breakpoints, the order in which coordinates enter, and the path length. It does not carry statements averaged over the response at a fixed parameter, such as post-selection intervals, because a fixed λ and a fixed c are different experiments.

🧪 Testing

Run the test suite with:

make test

or directly via pytest:

uv run pytest

🧹 Code Quality

Format and lint the code with:

make fmt

📖 Documentation

• Online Documentation
• Factor backend
• Notebooks
• Test report
• Coverage report
• The Critical Line Algorithm and the Constrained LASSO: One Curve, Two Literatures (Schmelzer and Hastie, 2026)

📚 Citing

If you use cvxcla, please cite the software through its Zenodo DOI (see CITATION.cff). If you use the CLA–LASSO correspondence, please also cite:

@misc{schmelzer2026onecurve,
  title         = {The Critical Line Algorithm and the Constrained {LASSO}:
                   One Curve, Two Literatures},
  author        = {Schmelzer, Thomas and Hastie, Trevor},
  year          = {2026},
  eprint        = {2609.25704},
  archivePrefix = {arXiv},
  primaryClass  = {stat.ME},
  url           = {https://arxiv.org/abs/2609.25704}
}

👥 Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

1. Fork the repository
2. Create your feature branch (git checkout -b feature/amazing-feature)
3. Run the tests to make sure everything works (make test)
4. Format your code (make fmt)
5. Commit your changes (git commit -m 'Add some amazing feature')
6. Push to the branch (git push origin feature/amazing-feature)
7. Open a Pull Request

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

🔍 Related Projects

• PyCLA by Philipp Schiele - A previous implementation of the Critical Line Algorithm in Python.

• CLA by Martin Dengler - The original implementation by David Bailey and Marcos Lopez de Prado.