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Detecting and Proving Infeasibility in Linear Programs

Exploring the cinematic intuition of Detecting and Proving Infeasibility in Linear Programs.

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The Formal Theorem

Consider the primal linear program in standard form: minimize cTx c^T x subject to Ax=b,x≥0 Ax = b, x \geq 0 . By Farkas' Lemma, the system Ax=b,x≥0 Ax = b, x \geq 0 is infeasible if and only if there exists a vector y∈Rm y \in \mathbb{R}^m such that the following certificate of infeasibility holds:
ATy≤0andbTy>0 A^T y \leq 0 \quad \text{and} \quad b^T y > 0

Analytical Intuition.

Imagine you are an architect trying to build a structure that satisfies a set of strict, competing constraints—your linear equations Ax=b Ax=b and non-negativity requirements x≥0 x \geq 0 . Infeasibility means your blueprint is logically impossible; the constraints contradict one another so deeply that no solution exists. How do we prove this 'impossibility' without checking infinite points? We use Farkas' Lemma. Think of y y as a set of 'weights' assigned to your constraints. If we can find a linear combination of our constraints ATy A^T y that pushes the left-hand side into the non-positive quadrant (≤0 \leq 0 ) while simultaneously pushing the requirement bTy b^T y to be strictly positive (>0 > 0 ), we have found a mathematical paradox. This vector y y acts as a 'witness'—it exposes a contradiction where you are forced to satisfy a requirement bTy>0 b^T y > 0 using resources that sum to less than zero. It is the ultimate diagnostic tool, turning the elusive search for a solution into the constructive search for a contradiction.
CAUTION

Institutional Warning.

Students often confuse 'infeasibility' with 'unboundedness'. Infeasibility means the feasible region is the empty set, whereas unboundedness implies the feasible region exists but extends to infinity, allowing the objective function to improve without limit.

Academic Inquiries.

01

How does the Simplex method detect infeasibility?

During the Phase I of the Two-Phase Simplex method, we introduce artificial variables. If the optimal objective value of the Phase I problem remains strictly positive, it proves that no feasible solution exists for the original constraints.

02

Is the certificate of infeasibility unique?

No. Farkas' Lemma guarantees the existence of at least one such vector y y , but there may be an entire polyhedral cone of such certificates, each providing a different perspective on why the system is inconsistent.

Standardized References.

  • Definitive Institutional SourceBertsimas, D., & Tsitsiklis, J. N., Introduction to Linear Optimization.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Detecting and Proving Infeasibility in Linear Programs: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/linear-and-integer-programming/detecting-and-proving-infeasibility-in-linear-programs

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