Proof that the Dual of a Dual LP is the Primal LP

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The Formal Theorem

Let

P

be a primal linear program in standard form:

\begin{array}{ll} \text{maximize} & c^T x \\ \text{subject to} & Ax \leq b \\ & x \geq 0 \end{array}

Its dual linear program

D

is:

\begin{array}{ll} \text{minimize} & b^T y \\ \text{subject to} & A^T y \geq c \\ & y \geq 0 \end{array}

Now, let

D'

be the dual of

D

. The proof establishes that

D'

is equivalent to the original primal program

P

(possibly after a transformation if

D

was not in standard form). Specifically, if we transform

D

to its standard form for dualization, its dual

D'

will be equivalent to

P

Analytical Intuition.

Imagine a master strategist, planning a grand campaign (the Primal LP). This strategist's plan has inherent strengths and weaknesses. Now, consider an adversary analyzing this plan from the opposite perspective, looking for vulnerabilities and counter-strategies (the Dual LP). This adversary, in turn, might have their own complex plan. The profound revelation is that when the *adversary* of the adversary devises *their* plan, it perfectly mirrors and reinforces the *original* master strategist's intentions. It's like looking into a mirror, then looking into the mirror of that mirror – you end up seeing the original image, with all its nuances and intentions perfectly preserved, albeit possibly from a different angle.

CAUTION

Institutional Warning.

The primary confusion arises from the necessary transformations required to put the dual LP into standard form before constructing *its* dual, which can obscure the direct correspondence to the original primal.

Academic Inquiries.

Why is it important to prove that the dual of the dual is the primal?

This property, known as symmetry, is fundamental to the duality theory of linear programming. It underpins many theoretical results, including the strong duality theorem, and provides a consistent framework for analyzing optimization problems from different perspectives.

Does this proof hold for all forms of linear programs (e.g., not just standard form)?

Yes, the principle holds. However, the proof becomes more involved as one needs to account for the specific transformations required to convert different forms (like equality constraints or unrestricted variables) into the standard form for dualization. The core relationship remains.

What if the primal LP has no feasible solution or an unbounded objective?

The symmetry still holds in a generalized sense. If the primal is infeasible, the dual is unbounded (or infeasible in a generalized context). If the primal is unbounded, the dual is infeasible. The dual of the dual will then reflect the original primal's status.

Standardized References.

Definitive Institutional SourceBertsimas, D., & Tsitsiklis, J. N. (1997). Introduction to Linear Optimization. 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 the Dual of a Dual LP is the Primal LP: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/linear-and-integer-programming/proof-that-the-dual-of-a-dual-lp-is-the-primal-lp

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