https://github.com/madvorak/duality

Duality theory in linear optimization and its extensions
https://github.com/madvorak/duality

lean4 linear-algebra linear-programming

Last synced: 6 months ago
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Duality theory in linear optimization and its extensions

• Host: GitHub
• URL: https://github.com/madvorak/duality
• Owner: madvorak
• License: apache-2.0
• Created: 2024-07-17T14:43:59.000Z (almost 2 years ago)
• Default Branch: main
• Last Pushed: 2025-01-20T10:26:41.000Z (over 1 year ago)
• Last Synced: 2025-01-20T10:31:12.761Z (over 1 year ago)
• Topics: lean4, linear-algebra, linear-programming
• Language: Lean
• Homepage:
• Size: 15.8 MB
• Stars: 6
• Watchers: 2
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# Duality theory in linear optimization and its extensions

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 in Lean 4.

Furthermore, we consider a linearly ordered field extended with two special elements denoted by $\bot$ and $\top$

where $\bot$ is below every element and $\top$ is above every element.

We define $\bot + a = \bot = a + \bot$ for all $a$ and we define $\top + b = \top = b + \top$ for all $b \neq \bot$.

Instead of multiplication, we define scalar action $c \bullet \bot = \bot$ for every $c \ge 0$ but we define

$d \bullet \top = \top$ only for $d > 0$ because $0 \bullet \top = 0$.

We extend certain Farkas-like theorems to a setting where coefficients are from an extended linearly ordered field.

![AI-generated image](nonLean/AI-generated.jpg)

[Technical report](https://arxiv.org/abs/2409.08119)

## Main corollaries

* [Farkas for equalities](https://github.com/madvorak/duality/blob/ea1887033d86c29999f64a6d5c056d2267f3814c/Duality/FarkasBasic.lean#L24)

* [Farkas for inequalities](https://github.com/madvorak/duality/blob/ea1887033d86c29999f64a6d5c056d2267f3814c/Duality/FarkasBasic.lean#L101)

* [Strong duality for standard LP](https://github.com/madvorak/duality/blob/ea1887033d86c29999f64a6d5c056d2267f3814c/Duality/LinearProgrammingB.lean#L203)

## Main results

* [Farkas-Bartl theorem](https://github.com/madvorak/duality/blob/ea1887033d86c29999f64a6d5c056d2267f3814c/Duality/FarkasBartl.lean#L216)

* [Extended Farkas theorem](https://github.com/madvorak/duality/blob/ea1887033d86c29999f64a6d5c056d2267f3814c/Duality/FarkasSpecial.lean#L284)

* [Strong duality for extended LP](https://github.com/madvorak/duality/blob/ea1887033d86c29999f64a6d5c056d2267f3814c/Duality/LinearProgramming.lean#L1083)