What does this research mean for the field?

Key foundational theorems of linear programming, including weak duality and complementary slackness, can be fully formalized and machine-verified using the Lean 4 interactive theorem prover. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To comprehensively analyze Smale's 9th Problem regarding the existence of a strongly polynomial-time algorithm for linear programming and formally verify core foundational theorems in Lean 4.

August 30, 2026Open Access

Strongly Polynomial Bounds in Linear Programming and Smale's 9th Problem

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Authors

CNCharles EDOU NZE

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Implication

Theoretical analysis demonstrates core algorithmic barriers and bounds in linear programming, providing Lean 4 machine-verified proofs for fundamental duality theorems.

Key Points

To comprehensively analyze Smale's 9th Problem regarding the existence of a strongly polynomial-time algorithm for linear programming and formally verify core foundational theorems in Lean 4.
Rigorous mathematical exposition synthesizing primal-dual complementary slackness, Éva Tardos' subdeterminant theorem, the Vavasis-Ye layered least-squares method, and tropical geometry curvature barriers.
Interactive formal theorem verification using Lean 4 and Mathlib to formally check foundational linear programming identities and theorems without extra axioms or unproven placeholders.
Provided an algebraic proof showing LP is solvable in O(poly(m, n, log Δ)) time independent of the bit-lengths of vectors b and c when subdeterminants are bounded by Δ.
Demonstrated that the Vavasis-Ye layered least-squares path-following method terminates in O(n^3.5 log(χ̄(A) + n)) iterations in ill-conditioned regimes.
Achieved 100% machine-checked formal verification in Lean 4 with zero axioms, zero linter warnings, and zero sorry placeholders for the duality gap identity, weak duality, complementary slackness, and unimodular integer preservation.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a93f0f56c1a8fb52e79d8cc https://doi.org/10.5281/zenodo.22148792

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Implication

Theoretical analysis demonstrates core algorithmic barriers and bounds in linear programming, providing Lean 4 machine-verified proofs for fundamental duality theorems.

Strongly Polynomial Bounds in Linear Programming and Smale's 9th Problem

Discussion

Member takes

Implication

Key Points

Cite This Study

Strongly Polynomial Bounds in Linear Programming and Smale's 9th Problem

Discussion

Member takes

Implication

Key Points

Cite This Study

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