[tschm]

Restructures the constrained-LASSO note for readability, addresses every point of an editor's review for the Journal of Reproducible Statistics, and adds a CI workflow for the R step. (#67 is merged; this is based on main.)

Section 1: a result, not a walked-back prediction

"What this note checks" said the CLA "should return the knots … with no change to its code", then that this "was not quite met, in two ways". It is now "What this note shows": a CLA written for portfolios, with a thin layer (two encodings and a relaxation), gives exact constrained-LASSO paths, and with p > n it does so with no ridge, which the other open path algorithms for general constraints don't. The two CLA defects are still stated and point to Appendix C.

Section 6: four parts instead of ten special cases

1. Degenerate starts, and the relaxation that resolves them.
2. Starts that need no relaxation: rows slack at β = 0, flat starts, rows that exclude β = 0.
3. More predictors than observations (unchanged).
4. What stays out of reach: numerically singular free blocks, and a relaxation that no retry certifies.

Appendices

• A. The monotone fit: the knots move linearly in ε.
• B. One encoding or two, with the encodings table.
• C. What the checks did for the CLA, with the portfolio table.
• D. The detail on Gaines et al.: the port, the started problems, and why its shipped start is off the path. Section 5 now summarises it in one paragraph.

No text was dropped; everything moved is in an appendix.

Hand-typed numbers now from the script

Wide's 53 knots; where lars_path's positive mode really ends (λ = 30.65) and its KKT violation at λ = 0 (3.2e-2); Figure 2's 13 knots, 2.3e-13 and ‖β(0)‖₁ = 3460. They go to figures/evidence_numbers.tex, and test_lars_positive_end checks the claim.

Review fixes

1. Inequality rows in the CLA (Section 2). Problem (2) now has Cx ≤ h, which Proposition 1, the code and the evidence already passed. Active rows join the bordered system as equalities. A slack row joins when its slack reaches zero; an active row leaves when its multiplier does.

2. The relaxation, specified (Section 6). Lemma 1 states when the ε/2ε extrapolation is exact (both runs take the same sequence of active sets) and proves it; the certificate is what the result rests on. The parameters are as cvxcla runs them: ε = ρ‖c‖∞ / max G_jj with ρ = 1e-6, then 1e-7 and 1e-8; d from a fixed seed with signs the bounds allow, scaled so C_r d ≤ ½; acceptance at 1e-9 relative. Appendix A is now an illustration, not a proof.

3. comlasso, run (Section 5). R/comlasso_check.R fetches comlasso 0.2 from jenjong/ComLasso at a pinned commit, checks its SHA-256, and loads the package's code unchanged (R 4 won't install a package built for R 3.2). It traces the same knots as the CLA on Zero-sum, sCD14 and Wide, zero-sum (3 seeds):

   problem	gap (relative to the largest coefficient)
   Zero-sum, sCD14	≤ 4.1e-15
   Wide, zero-sum (p > n)	≤ 5.8e-12

   On two Wide seeds comlasso steps to λ ≈ 1e-11 before recording λ = 0; cvxcla counts that as λ = 0. The script fails on a gap above 1e-9 or a knot mismatch.

   make figures now runs R checks marked # run: after figures. compile.yml installs R 4.6.1 and the packages a check names on # needs: (Ubuntu libraries on # apt:) from the CRAN snapshot r-check.yml already pins. This is the first CI run of that step; actionlint wasn't available locally.

Minor review points (4–12)

• Abstract: now mentions the relaxation layer, the agreement with comlasso, and the one declined problem.
• Numbers from the script:
  • Pinned's first knot (4053), the largest interior-point gap of Table 2, and the p = 500 cross-check in Scale are now macros.
  • The cost ratios are said to be measured once by timings.
• Uniqueness: stated as an assumption when there are constraints or p > n; the checks test optimality, not uniqueness.
• Table 6: the caption names its seed.
• Figure 3: the 8 largest genera are in colour and the rest in grey, with interior-point checks at every third midpoint.
• Layout:
  • Appendix tables sit with their text; Gaines comes before the portfolio defects.
  • The bibliography is ragged-right, and the build has no LaTeX warnings.
• Related work follows the encodings list, under its own heading.

CI: comlasso-check

A new workflow, modelled on r-check.yml, runs on PRs touching preprints/constrained_lasso/**, and can be started by hand. It:

1. fetches sCD14;
2. writes the saved paths with make_figures.py export, a new mode that takes about 1.5 s;
3. installs R 4.6.1 and Rglpk 0.6.5.1 (pinned) from the CRAN snapshot;
4. runs R/comlasso_check.R and uploads figures/comlasso.tex.

Its first run passed: the same knots on all five problems, gaps ≤ 2.5e-12. compile.yml also runs the check inside make figures, and passed with it.

Checks

• make_figures.py test: 17 passed.
• CI: compile (both preprints) and comlasso-check green.
• R/comlasso_check.R: passes locally (R 4.6.1, Rglpk 0.6.5.1, GLPK 5.0).
• Builds to 17 pages (main text about 12, then references and the appendices), with no overfull lines and no undefined references.

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