# Standard form of a linear program

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0.

mko-linear-program-standard-form · definition · v0.10.0



## Mathematical statement

A is an exact rational m-by-n matrix with matching b,c and decision vector x. Qualify dimensions and the chosen sign convention first. Nonnegative slack s=b−A*x converts A*x≤b to A*x+s=b, preserving feasibility and c^T*x in both directions between original and slack points. A free variable splits as x_plus−x_minus with both parts nonnegative; negate a ≥ constraint, replace equality by opposite ≤ rows, and negate a minimization objective. These conversions do not prove feasibility, boundedness or an optimal solution, and another standard-form convention is not silently substituted.





## Representative formula (exact TeX)

L(A,b,c)=z





## Assumptions


- Matrices, vectors and points are bounded exact rationals.
- Statements and outputs consistently use maximize/≤/nonnegative.





## Symbol scope


- L — linear-program standard-form representation
- A — constraint matrix
- b — right-hand-side vector
- c — objective vector
- z — objective value





## Intuition, definition, and misconceptions

A single convention avoids standard-form ambiguity.

Slack conversion includes both directions of the point correspondence.

Free variables, ≥, equality and minimization have separately stated conversions.

Do not mix objective/inequality conventions or mistake a representation conversion for feasibility or optimality.





## Adopted derivation guide

[The two-way point correspondence for slack variables](https://ocme.evemisslab.com/en/derivations/derivation-mko-linear-program-standard-form-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Linear programming](https://ocme.evemisslab.com/en/mko/mko-linear-programming/) — Fix the objective and constraints.
- [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/mko/mko-matrix/) — Use matrix-vector multiplication.
- [Linear systems in two variables and degenerate constraints](https://ocme.evemisslab.com/en/mko/mko-system-linear-equations/) — Slack variables give an equality form.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Inequalities are componentwise.





## Verification boundary

Only representation and point correspondences are certified, not feasibility, boundedness, optimality or duality. A free-variable split is nonunique even though the slack map has a unique slack for each x.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add linear-programming duality.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/linear-program-standard-form.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-linear-program-standard-form/index.md)