Fix a bug in dual variables and reduced costs when we add implied bounds on free variables

[chris-maes]

When -inf <= x_j <= inf and we add implied bounds on x_j, we will compute a dual solution (u, w) that satisfies

A^T u + w = c + Qx

but w_j could be nonzero. When x_j is free, we need to enforce that the corresponding reduced cost is zero. We do this by transforming (u, w) into (y, z) such that A^T y + z = c + Qx with z_j = 0 for all j such that -inf < x_j < inf.

We also add a unit test on a small QP to confirm the fix.

Fixes #1119

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[chris-maes]

/ok to test 23789ff

[chris-maes]

/ok to test 06eb7d2

[coderabbitai]

📝 Walkthrough

Walkthrough

This PR extends presolve to track which constraints imply finite bounds on formerly-free variables, then applies a post-solve dual correction during solution uncrushing. Presolve records constraint indices and coefficients for each bounded variable, and uncrush uses these to adjust dual variables and reduced costs via the original constraint matrix.

Changes

Barrier dual correction for bounded free variables

Layer / File(s)	Summary
Data structures for bounded free variable tracking cpp/src/dual_simplex/presolve.hpp	Introduces bounded_free_var_t<i_t, f_t> to store variable index, originating constraint index, and coefficient. Extends presolve_info_t<i_t, f_t> with bounded_free_variables member. Updates uncrush_solution declaration to accept original_problem parameter.
Presolve tracking of bound-originating constraints cpp/src/dual_simplex/presolve.cpp	Adds per-variable buffers in presolve() to record constraint and coefficient for each finite bound acquired by a free variable. After bound selection, populates presolve_info.bounded_free_variables with variable, constraint, and coefficient tuples for each bounded free variable.
Uncrush dual correction for bounded free variables cpp/src/dual_simplex/presolve.cpp	Updates uncrush_solution() to accept original problem matrix. Implements post-solve correction block that computes du = w_j / coefficient for each bounded free variable, updates dual variable input_y[constraint] += du, and adjusts reduced costs via original_problem.A. Updates template instantiation signature.
Solver integration of uncrush with original problem cpp/src/dual_simplex/solve.cpp	Both solve_linear_program_with_advanced_basis and solve_linear_program_with_barrier now pass original_lp when calling uncrush_solution().
Test infrastructure and free-variable dual correction validation cpp/tests/dual_simplex/unit_tests/solve.cpp, cpp/tests/dual_simplex/unit_tests/solve_barrier.cu	Existing tests initialize cuopt::init_logger_t for diagnostics. New TEST(barrier, min_x_squared_free_variable_dual_correction) loads an MPS dataset, solves via PDLP, and validates primal solution, dual solution, and reduced-cost correctness under the dual correction mechanism.

🎯 3 (Moderate) | ⏱️ ~20 minutes

🚥 Pre-merge checks | ✅ 4 | ❌ 1

❌ Failed checks (1 warning)

Check name	Status	Explanation	Resolution
Docstring Coverage	⚠️ Warning	Docstring coverage is 0.00% which is insufficient. The required threshold is 80.00%.	Write docstrings for the functions missing them to satisfy the coverage threshold.

✅ Passed checks (4 passed)

Check name	Status	Explanation
Title check	✅ Passed	The title accurately summarizes the main change: fixing a bug related to dual variables and reduced costs when implied bounds are added on free variables.
Linked Issues check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Out of Scope Changes check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Description check	✅ Passed	The PR description accurately describes the mathematical problem being solved and explains the dual correction strategy for free variables with implied bounds.

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✨ Finishing Touches

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Actionable comments posted: 1

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Inline comments:
In `@cpp/src/dual_simplex/presolve.cpp`:
- Around line 1758-1759: The dual correction unconditionally computes du = w_j /
bfv.coefficient which can produce Inf/NaN when bfv.coefficient is zero or
near-zero; in the block around du, guard the division by checking
fabs(bfv.coefficient) against a small epsilon (e.g. machine or
domain-appropriate epsilon) and only compute and apply input_y[bfv.constraint]
+= du when the coefficient is safe; if the coefficient is too small, skip the
update or handle it deterministically (clamp, set du = 0, or record/log the
event) to avoid division-by-zero and maintain numerical stability in the dual
correction path.

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📥 Commits

Reviewing files that changed from the base of the PR and between 6f99a42 and 23789ff.

📒 Files selected for processing (5)

• cpp/src/dual_simplex/presolve.cpp
• cpp/src/dual_simplex/presolve.hpp
• cpp/src/dual_simplex/solve.cpp
• cpp/tests/dual_simplex/unit_tests/solve.cpp
• cpp/tests/dual_simplex/unit_tests/solve_barrier.cu