mean_variance_solvers

LaTeX source for a set of companion papers on fast, matrix-free solvers for the long-only minimum-variance / mean-variance portfolio problem, by Thomas Schmelzer (Jebel Quant Research), Martin Stoll (TU Chemnitz), and Michael Wolf (University of Zurich / ADIA Lab):

• Matrix-Free Methods for Long-Only Portfolio Optimization — under matrix_free/. Casts the long-only minimum-variance problem so that covariance shrinkage acts as a preconditioner, and solves it with matrix-free iterative methods that never form the dense covariance matrix.
• From Marchenko–Pastur to Woodbury: Direct Solvers for Long-Only Mean-Variance Portfolios — under rmt/. Uses random-matrix theory (Marchenko–Pastur) to motivate a low-rank-plus-diagonal covariance model that a Woodbury identity turns into a fast direct solver.
• Non-Negative Conjugate Gradients — under non_negative_cg/, published as arXiv:2607.22121 (Schmelzer and Stoll, 24 July 2026). A domain-neutral technical note that isolates the computational kernel shared by the two papers above: solving the bound-constrained SPD quadratic min_{x≥0} ½xᵀAx − bᵀx (and its least-squares / equality-augmented variants) by wrapping matrix-free conjugate gradients in a primal-dual active-set loop, with an unconditional finite-termination guarantee and an operator abstraction that admits dense, Gram, factor/Woodbury, and regularised backends. It contains no finance material; the other two papers are, in its terms, two backends of one solver.
• Goldfarb–Idnani Revisited: Invariants, Certificates, and the Limits of Guessing — under quadprog/. A derivation of the Goldfarb–Idnani dual active-set method in the form the cvx-quadprog package implements it: the two invariants the solver's single carried matrix satisfies, the closed forms that make the iterates independent of how that matrix is represented, the exact objective increment, and the Farkas certificate hiding inside the infeasibility verdict. Also domain-neutral, and the general-constraint counterpart to the note above: where that one solves min_{x≥0} ½xᵀAx − bᵀx and keeps a finite-termination guarantee, this one handles C^T x ≥ b and shows precisely which structure is lost in doing so. The companion implementation paper lives in that package's own repository.

All four papers share one bibliography (matrix_free/bib/refs.bib). The two finance papers draw every figure and table from the same numerical experiments in experiment/; the non-negativity note is self-contained and draws its synthetic-study figures and tables from its own non_negative_cg/experiment/ (NumPy only, no finance data).

Building

A Makefile at the repository root drives everything; run make (or make help) for the list of targets:

make compile   # build every paper (cg_paper.pdf, rmt_paper.pdf, nncg_paper.pdf)
make figures   # regenerate every paper's figures and tables (finance experiment + nncg study)
make arxiv     # assemble a self-contained arXiv source tarball per paper
make clean     # remove both papers' LaTeX build artifacts (keeps the PDFs)

make compile runs pdflatex → bibtex → pdflatex → pdflatex per paper: the first pass records the citations, BibTeX turns them into a formatted bibliography, and the final two passes fold it in and resolve all cross-references. The minimum-variance paper uses the standard siam bibliography style; the RMT paper uses plain. Both ship with any full TeX Live install, so no class or style files need to be vendored.

make figures delegates to experiment/, whose scripts are self-contained PEP 723 programs run with uv; their pinned dependencies include the fast-minimum-variance package, resolved directly from its git repository. The committed figure PDFs and table .tex files mean the papers compile without ever running the experiment.

Layout

Makefile                 root entry point: delegates to the per-paper Makefiles
common.mk                shared build logic (compile / arxiv / clean)
matrix_free/
  Makefile               builds cg_paper.pdf (include ../common.mk)
  cg_paper.tex       main file: preamble + \input of the sections
  sections/              one .tex per section (s0_abstract … s8_conclusions)
  bib/refs.bib           shared BibTeX database
  siam/                  vendored SIAM class/style (reference copy)
  graphs -> ../experiment/graphs    figure PDFs (symlink)
  tables -> ../experiment/tables    table .tex fragments (symlink)
rmt/
  Makefile               builds rmt_paper.pdf (cites ../matrix_free/bib/refs.bib)
  rmt_paper.tex          main file: preamble + \input of the sections
  sections/              one .tex per section (s0_abstract … s8_conclusions)
  graphs -> ../experiment/graphs
  tables -> ../experiment/tables
non_negative_cg/
  Makefile               builds nncg_paper.pdf (cites ../matrix_free/bib/refs.bib)
  nncg_paper.tex         main file: preamble + \input of the sections
  sections/              one .tex per section (s0_abstract … s8_conclusions)
  experiment/            self-contained synthetic study (nncg.py + experiment_nncg.py,
                         NumPy only) and an external-solver benchmark
                         (experiment_nncg_bench.py; SciPy + Clarabel); the algorithms
                         are released as the pip package
                         https://github.com/Jebel-Quant/nncg
  graphs/                generated figure PDFs (committed; `make figures`)
  tables/                generated table + \newcommand fragments (committed)
experiment/              numerical experiments that generate graphs/ and tables/
  experiment*.py         PEP 723 scripts (real-data, synthetic, OOS, RMT)
  fetch_*.py             download the raw S&P 500 / FTSE 100 return data
  data/                  committed return data (*.parquet)
  graphs/                generated figure PDFs (committed)
  tables/                generated table .tex fragments (committed)
  util/                  shared table/timing helpers

Adding a new paper is a matter of dropping in a folder with a Makefile that sets DOC and include ../common.mk; the root Makefile discovers it automatically (the experiment/ folder is excluded).

Continuous integration

GitHub Actions workflows (.github/workflows/):

• build — compiles both papers (make compile) and publishes cg_paper.pdf and rmt_paper.pdf to the pdf branch; on tags it also attaches the PDFs and the arXiv source tarballs to a GitHub release.
• arxiv — assembles a self-contained arXiv source tarball for each paper (make arxiv) and publishes both to the arxiv branch.

Branches

• main — primary development branch.
• pdf — orphan branch holding the compiled PDFs; never merged into main.
• arxiv — orphan branch for arXiv submission artifacts; never merged into main.