Proof that if the Primal is Unbounded, the Dual is Infeasible (and vice-versa)

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The Formal Theorem

Consider a Linear Programming problem in standard form: Primal:

\min c^T x

subject to

Ax \ge b

x \ge 0

. Its dual is: Dual:

\max b^T y

subject to

A^T y \le c

y \ge 0

. If the primal problem is unbounded, then the dual problem is infeasible. Conversely, if the dual problem is infeasible, then the primal problem is unbounded. Mathematically, this can be stated as:

\text{Primal Unbounded} \iff \text{Dual Infeasible}

Analytical Intuition.

Imagine two warring kingdoms, Primal and Dual, vying for dominance. Primal seeks to expand its territory infinitely (unbounded objective value) by conquering land

Ax \ge b

with limited resources

x \ge 0

. Its opponent, Dual, tries to build an impenetrable fortress (infeasible dual) by setting up trade routes

A^T y \le c

with limited guards

y \ge 0

. If Primal's expansion is unstoppable, it means Dual's fortress defenses are fundamentally flawed and cannot contain Primal's growth, thus Dual must be empty (infeasible). Conversely, if Dual cannot even establish a viable defense, its structure must be inherently broken, implying Primal has no limits to its relentless advance.

CAUTION

Institutional Warning.

The subtle interplay between unboundedness and infeasibility arises from the nature of duality. An unbounded primal implies no feasible solution can 'contain' its growth, while an infeasible dual means no feasible solution 'bounds' its potential.

Academic Inquiries.

What does it mean for a linear program to be 'unbounded'?

A linear program is unbounded if its objective function can attain arbitrarily large (or small, for minimization) values while still satisfying all constraints. This implies there are feasible solutions that can be improved indefinitely.

What does it mean for a linear program to be 'infeasible'?

A linear program is infeasible if there is no set of decision variables that can simultaneously satisfy all the constraints of the problem. The feasible region is empty.

Why is the proof of this theorem important?

This theorem is a cornerstone of duality theory in linear programming. It provides a powerful link between the solvability of a primal problem and its dual, guiding algorithmic design and theoretical understanding. It implies that if a problem has a solution, either the primal is bounded and has an optimal solution, or it's infeasible.

Can a problem be both unbounded and infeasible?

No. According to the strong duality theorem and its consequences like this one, a primal problem can be: (1) bounded and have an optimal solution, (2) unbounded and have an infeasible dual, or (3) infeasible and have an unbounded dual. It cannot be simultaneously unbounded and infeasible.

Standardized References.

Definitive Institutional SourceBertsekas, D. P. (1999). Nonlinear programming. Athena Scientific.

Foundational

The Convexity of the Feasible Region of a Linear Program

Exploring the cinematic intuition of The Convexity of the Feasible Region of a Linear Program.

Intermediate

The Fundamental Theorem of Linear Programming: Existence of an Optimal Extreme Point Solution

Exploring the cinematic intuition of The Fundamental Theorem of Linear Programming: Existence of an Optimal Extreme Point Solution.

Intermediate

Equivalence of Basic Feasible Solutions and Extreme Points

Exploring the cinematic intuition of Equivalence of Basic Feasible Solutions and Extreme Points.

Advanced

Characterization of Unboundedness in Linear Programming

Exploring the cinematic intuition of Characterization of Unboundedness in Linear Programming.

Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof that if the Primal is Unbounded, the Dual is Infeasible (and vice-versa): Visual Proof & Intuition. Retrieved from https://nicefa.org/library/linear-and-integer-programming/proof-that-if-the-primal-is-unbounded--the-dual-is-infeasible--and-vice-versa-

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