[tschm]

With more predictors than observations, the path algorithms for general constraints (Gaines et al., CLASSO, c-lasso) add a ridge to XᵀX; the CLA needs none. The note claimed this on a single problem. This PR backs it with evidence and measures what the ridge costs.

New evidence

• Table 5 (new): Wide (n = 40, p = 100, three seeds) under nonnegativity, a box, two binding inequality rows, a nonhomogeneous row, and a zero sum with signs or caps.

  Wide with	knots	gap (IPM)	KKT
  nonnegative	56–100	2.2e-9	2.5e-15
  box	92–228	2.6e-8	8.2e-16
  inequality rows	54–62	8.0e-8	1.8e-14
  pinned	61–73	2.9e-8	1.8e-15
  zero-sum, signs	44–56	4.7e-9	2.7e-13
  zero-sum, caps	88–245	8.7e-7	1.6e-13

  None declined; KKT is checked along whole segments.

• What the ridge costs, measured: c-lasso on Wide, zero-sum, run down to 1e-9·λ_max so it reaches the end of the path. As shipped it lies 0.41–0.58 × the largest coefficient from the exact path, against 6.1e-5 when n ≥ p. With the ridge at 1e-12 the gap is 5.8e-12, so the ridge is the whole difference: with p > n it decides where the path ends.

• CLASSO's own figures: its Table 3 reports objectives up to 4e-2 above the unregularised ones at its default ridge.

Claim narrowed

• comlasso (Jeon et al., CSDA 148, 2020, doi:10.1016/j.csda.2020.106958) traces the exact zero-sum path with p > n and no ridge. It is now cited, and the claim is limited to general constraints (bounds, signs, inequality rows).
• Monotone fits with p > n are not claimed: one seed in three is declined.

Text

The abstract, Section 1 (new paragraph), Section 5 (new paragraph, Table 5, c-lasso result), Section 6 and the conclusion. The encodings and portfolio tables become Tables 6 and 7.

Code

• make_figures.py: wide_problems, stress() takes the problem set, Table 5 and its numbers, test_wide, and the p > n paths saved for the c-lasso check.
• classo_check.py: runs the p > n paths with a small lamin and writes four new numbers. The two existing c-lasso numbers are unchanged.

One tolerance is looser: test_wide allows the IPM objective to beat the path by 1e-11 (Table 4 uses 1e-12). The IPM gets ahead only near the end of a path, where the objective has fallen to about 1e-4 of its value at β = 0. Measured against that starting value, the 4.9e-12 excess is about 1e-14, which is rounding.

Checks

• make_figures.py test: 15 passed.
• Every existing table and number is regenerated unchanged.
• The note compiles to 16 pages, no overfull lines.
• Independent of constrained_lasso: say where the CLA alone falls short #65; the two touch different lines.

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