Updated matrix F in Palma Ratio Linearization

[samtalki]

Matrix F Corrections for Palma Ratio Linearization

This PR suggests some changes to the constraint matrix F and related components in the Palma ratio linearization implementation. It attempts to resolve the bug mentioned in #3

I tried to align things with the mathematical formulation on the Overleaf. I used Claude Code Pro with the Opus 4.5 model to try to help.

Possible issues

I believe there are three possible issues in the current implementation:

1. Indexing Inconsistency: Mixed row-major and column-major vectorization conventions
2. McCormick Envelope RHS Errors: Possibly incorrect right-hand side values for McCormick constraints
3. Undefined Variable References: A_long used undefined variable A

---

Issue 1: Indexing Inconsistency

Problem

The original code mixed two different vectorization conventions for the permutation matrix:

Component	Original Indexing
A, AT, T	Row-major: k = (i-1)*n + j
A_row, A_col	Column-major: k = i + (j-1)*n

This caused the sorting constraint (using T) to operate inconsistently with the permutation constraints (using A_row, A_col).

Example for n=3

For a permutation matrix entry a_{i,j}:

• Row-major: vec(a) = [a_{11}, a_{12}, a_{13}, a_{21}, a_{22}, a_{23}, a_{31}, a_{32}, a_{33}]
• Column-major: vec(a) = [a_{11}, a_{21}, a_{31}, a_{12}, a_{22}, a_{32}, a_{13}, a_{23}, a_{33}]

Suggested Fix

Standardized all matrices to use column-major indexing, as is standard in Julia/MATLAB convention:

# Column-major indexing: a[k] = a_{i,j} where k = i + (j-1)*n
A_row = zeros(n, n^2)
for i in 1:n
    for j in 1:n
        idx = i + (j-1)*n
        A_row[i, idx] = 1.0
    end
end

The sorting matrix T was also updated to use column-major and now has dimension (n-1) × n² instead of n × n²:

T = zeros(n-1, n^2)
for k in 1:n-1
    for j in 1:n
        idx_k = k + (j-1)*n
        idx_k1 = (k+1) + (j-1)*n
        T[k, idx_k] = 1.0
        T[k, idx_k1] = -1.0
    end
end

---

Issue 2: McCormick Envelope RHS Errors

Reference (Overleaf main.tex equations 19-22)

The McCormick envelopes for the bilinear term u_{ij} = a_{ij} · x_j where a_{ij} ∈ {0,1} and x_j ∈ [0, P_j]:

Eq.	Constraint	Description
(19)	u_{ij} ≥ 0	Lower bound from a=0
(20)	u_{ij} ≥ x̄_j · a_{ij} + x_j - x̄_j	Lower bound from a=1
(21)	u_{ij} ≤ x̄_j · a_{ij}	Upper bound from x=x̄
(22)	u_{ij} ≤ x_j	Upper bound from a=0

Converting to F·y ≤ g·σ form

Envelope	F row	Correct RHS
(19)	-I · u ≤ 0	zeros(n²)
(20)	-I · u + A_P · a + x_j · x ≤ P	P_out
(21)	I · u - A_P · a ≤ 0	zeros(n²)
(22)	I · u - x_j · x ≤ 0	zeros(n²)

Original Code

g = [ones(n), -ones(n), ones(n), -ones(n), zeros(n),
     zeros(n^2),   # envelope 1: correct
     -P_out,       # envelope 3: seems wrong (should be zeros?)
     zeros(n^2),   # envelope 2: seems wrong (should be P_out?)
     zeros(n^2)]   # envelope 4: correct

Fixed Code

g = [ones(n), -ones(n), ones(n), -ones(n), zeros(n-1),
     zeros(n^2),   # envelope 1: -u ≤ 0
     zeros(n^2),   # envelope 3: u - a·P ≤ 0 (Proposed Fix)
     P_out,        # envelope 2: -u + a·P + x ≤ P (Proposed Fix)
     zeros(n^2)]   # envelope 4: u - x ≤ 0

---

Issue 3: A_long Variable Reference

Problem

A_long = [A zeros(n,n^2) zeros(n,2*n)]  # A was undefined/row-major

Fix

A_long = [A_row zeros(n,n^2) zeros(n,2*n)]  # Uses column-major A_row

---

Files Modified

1. src/implementation/palma_relaxation.jl

   • lin_palma() function (lines 14-145)
   • lin_palma_w_grad_input() function (lines 160-348)

2. script/relaxed_palma_ratio.jl

   • Test script (lines 312-470)

---

Dimension Changes

Due to the T matrix correction, the constraint system dimensions changed:

Component	Original	Fixed
T matrix	n × n²	(n-1) × n²
g vector sorting block	zeros(n)	zeros(n-1)
Total F rows	4n + n + 4n²	4n + (n-1) + 4n²

---

Added Verification

Dimension assertions were added to catch future dimension mismatches:

@assert size(F, 2) == length(y) "F columns must match y length"
@assert size(F, 1) == length(g) "F rows must match g length"

---

Trying to align with overleaf formulation

The corrected implementation now matches the Overleaf formulation (assuming it is still accurate):

1. Equations (5)-(6): Doubly stochastic constraints A·1 = 1, Aᵀ·1 = 1 are correctly implemented via A_row and A_col

2. Equation (8): Sorting constraint S_i ≤ S_{i+1} is implemented via T · x_hat ≤ 0

3. Equations (19)-(22): McCormick envelopes now have correct RHS values

4. Equations (31)-(34): Charnes-Cooper transformation F·y ≤ g·σ with the currently written constraint structure

---

Testing

I created a proof-of-concept test script test/test_matrix_f.jl to verify:

1. Matrix dimensions are consistent
2. Permutation constraints are satisfied
3. Sorting produces ascending order
4. McCormick envelopes are correctly bounded
5. Full constraint system F·y ≤ g is satisfied

Test Results

============================================================
Testing Matrix F Construction for Palma Ratio Linearization
============================================================

Test parameters: n = 3, P = [1.0, 2.0, 3.0]

--- Test 1: Dimension Consistency ---
Test Summary:         | Pass  Total
Dimension Consistency |    8      8

--- Test 2: Permutation Constraints ---
Identity permutation: Row sums = [1,1,1], Column sums = [1,1,1] ✓
Swap permutation: Row sums = [1,1,1], Column sums = [1,1,1] ✓
Test Summary:           | Pass  Total
Permutation Constraints |    2      2

--- Test 3: Sorting Constraint ---
Unsorted x: [3.0, 1.0, 2.0] → Sorted: [1.0, 2.0, 3.0]
T * x_hat = [-1.0, -1.0] (≤ 0) ✓
Test Summary:      | Pass  Total
Sorting Constraint |    2      2

--- Test 4: McCormick Envelope Bounds ---
For binary a ∈ {0,1}, bounds are tight at true value u = a·x ✓
Test Summary:       | Pass  Total
McCormick Envelopes |   12     12

--- Test 5: Full Constraint System ---
Max constraint violation: 0.0 ✓
All McCormick envelopes satisfied ✓
Test Summary:          | Pass  Total
Full Constraint System |   10     10

============================================================
All tests passed!
============================================================

Run the test with:

julia --project=. test/test_matrix_f.jl

[Copilot]

Pull request overview

This PR addresses three issues in the Palma ratio linearization implementation: standardizing vectorization indexing to column-major convention, correcting McCormick envelope RHS values, and fixing undefined variable references. The changes align the implementation with the mathematical formulation documented in Overleaf.

• Standardized all matrix indexing to column-major convention (k = i + (j-1)*n) for consistency
• Fixed McCormick envelope RHS values in the constraint vector g (swapped envelope 2 and 3 values)
• Corrected sorting constraint matrix T to have (n-1) rows instead of n rows

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 7 comments.

File	Description
test/test_matrix_f.jl	New comprehensive test file verifying matrix dimensions, permutation constraints, sorting behavior, McCormick envelopes, and full constraint system
src/implementation/palma_relaxation.jl	Updated both lin_palma() and lin_palma_w_grad_input() functions with column-major indexing, corrected T matrix dimensions, fixed McCormick RHS values, and added dimension assertions
script/relaxed_palma_ratio.jl	Applied same corrections as implementation file to maintain consistency in the test script

---

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[samtalki]

From the @copilot review, it looks like there was an inconsistency in -T vs T

• lin_palma() used T, but lin_palma_w_grad_input() and relaxed_palma_ratio.jl used -T (incorrect).
• In Overleaf T computes sorted[k] - sorted[k+1]. For ascending order (from eq 8: S_i ≤ S_{i+1}), we need T * x_hat ≤ 0, whereas -T would enforce descending order. I switched to ascending and used T*x_hat <= 0, let me know if this isn't what you wanted
• Changed -T to T in two locations.

@copilot does this resolve your concern?

[Copilot]

@samtalki I've opened a new pull request, #5, to work on those changes. Once the pull request is ready, I'll request review from you.

[awest32]

The functionality seems to be here. Thank you so much for doing this!