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Duality theory in linear optimization and its extensions -- formally verified (2409.08119v1)

Published 12 Sep 2024 in math.OC and cs.LO

Abstract: Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take ``infinite values''.

Summary

  • The paper rigorously verifies classical results such as Farkas’ lemma and the Strong Duality Theorem using Lean 4.
  • It extends duality theory to include infinite coefficients, broadening its applicability to discrete optimization problems.
  • The formal methods enhance algorithmic reliability and provide a solid foundation for advanced optimization techniques in AI.

Duality Theory in Linear Optimization and Its Extensions: Formally Verified

In the exploration of duality theory in linear programming, the paper "Duality theory in linear optimization and its extensions: formally verified" offers substantial contributions to our understanding and formal verification of several crucial theorems in this domain. Authored by Martin Dvorak and Vladimir Kolmogorov, this paper rigorously investigates and extends core principles of linear optimization through the use of Lean 4, a proof assistant software.

Overview and Contributions

The paper primarily focuses on three key areas:

  1. Formal Verification of Classical Results:
    • The Farkas' Lemma and its variants, including equality and inequality forms, are among the foundational results in linear programming. Existing versions in the literature have been rigorously proven using formal methods in Lean 4.
    • The Strong Duality Theorem for standard linear programs, which establishes a relationship between the primal and dual optimization problems, is also formally verified.
  2. Extensions to Infinite Coefficients:
    • The paper extends duality theory to scenarios where some coefficients are allowed to take "infinite values". This theoretical expansion is particularly significant for applications in discrete optimization problems with hard constraints.
    • A novel theorem analogous to Farkas' lemma for matrices with elements from an extended ordered field FF_{\infty} is introduced and proven.
  3. Generalizations and Expanded Applications:
    • A further generalization of the classical Farkas' lemma to non-commutative and infinitely indexed structures is discussed, broadening the applicability of duality theory in more complex algebraic systems.

Detailed Contributions

Classical Farkas' Lemma and Its Variants

In the classical setting of linear programming over linearly ordered fields, two well-known theorems are verified:

  • Equality Form (Farkas' Lemma):

    For a matrix A and vector b, exactly one of the following holds:\text{For a matrix } A \text{ and vector } b, \text{ exactly one of the following holds:}

    • x0  s.t.  Ax=b\exists x \ge 0\; \text{s.t.}\; A \cdot x = b
    • y0  s.t.  ATy0  and  by<0\exists y \ge 0\; \text{s.t.}\; A^T \cdot y \ge 0 \; \text{and} \; b \cdot y < 0
  • Inequality Form:

    For a matrix A and vector b, exactly one of the following holds:\text{For a matrix } A \text{ and vector } b, \text{ exactly one of the following holds:}

    • x0  s.t.  Axb\exists x \ge 0 \;\text{s.t.}\; A \cdot x \le b
    • y0  s.t.  ATy0  and  by<0\exists y \ge 0 \;\text{s.t.}\; A^T \cdot y \ge 0 \; \text{and} \; b \cdot y < 0

These results provide the theoretical underpinning for the duality in linear programming.

Strong Duality Theorem

The paper restates and formally verifies the Strong Duality Theorem: min{cxx0    Axb}=min{byy0    ATyc}\min \{ c \cdot x \mid x \ge 0 \; \wedge \; A \cdot x \le b \} = -\min \{ b \cdot y \mid y \ge 0 \; \wedge \; -A^T \cdot y \le c \} implying that if either primal or dual has a feasible solution, their optima are negatives of each other.

Extensions to Infinite Coefficients

A significant novelty of this paper is the introduction of extended ordered fields FF_{\infty}, which allow for "infinite values" in the coefficients. The main theorem in this context is: For AFI×J and bFI, exactly one of the following holds:\text{For } A \in F_{\infty}^{I \times J} \text{ and } b \in F_{\infty}^{I}, \text{ exactly one of the following holds:}

  • x:JF+  s.t.  Axb\exists x : J \rightarrow F^+_{\infty} \; \text{s.t.} \; A \cdot x \le b
  • y:IF+  s.t.  ATy0  and  by<0\exists y : I \rightarrow F^+_{\infty} \; \text{s.t.} \; -A^T \cdot y \le 0 \; \text{and} \; b \cdot y < 0

This theoretical enlargement is notable as it aligns discrete optimization problems with hard constraints into the duality framework without separating hard and soft constraints.

Theoretical and Practical Implications

The formally verified results ensure mathematical rigor and eliminate ambiguities or errors in the manual proofs of these theorems. The extension to matrices with infinite coefficients simplifies practical problem formulations in discrete optimization where "hard" constraints need recognition. Additionally, this framework aids the formal development of algorithms and software in linear optimization.

Speculations on Future Developments in AI

Formal verification of mathematical theorems using tools like Lean 4 can greatly enhance the robustness and reliability of AI systems, particularly those involved in optimization tasks. By ensuring that the underlying mathematical principles are sound, AI algorithms can be made more trustworthy and their behaviors more predictable. The incorporation of extended ordered fields might lead to more sophisticated optimization algorithms capable of dealing with more complex constraint systems directly, further pushing the boundaries of what AI can solve efficiently.

Conclusion

"Duality theory in linear optimization and its extensions: formally verified" offers a meticulously detailed formal verification of fundamental linear optimization theorems and presents new theoretical advancements that accommodate infinite values in coefficients. This work not only reinforces trust in these critical results through rigorous formal methods but also sets the stage for future extensions and developments in optimization theory and its applications in computer science and AI. The use of Lean 4 demonstrates the power of formal tools in advancing mathematical rigor and paves the way for integrating such methodologies into broader scientific and engineering disciplines.