Proof that All Optimal Solutions to a Linear Program Form a Convex Set

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The Formal Theorem

Consider a linear program in standard form: minimize

c^T x

subject to

Ax = b

and

x \geq 0

. Let

S = \{ x \in \mathbb{R}^n : Ax = b, x \geq 0 \}

be the feasible region. Let

z^*

be the optimal objective value. The set of optimal solutions

X^* = \{ x \in S : c^T x = z^* \}

is a convex set. Specifically, for any two points

x_1, x_2 \in X^*

and any scalar

\lambda \in [0, 1]

, the convex combination

x_{\lambda} = \lambda x_1 + (1 - \lambda) x_2

satisfies

x_{\lambda} \in X^*

Analytical Intuition.

Imagine the feasible region

S

as a multi-faceted diamond suspended in space, defined by the intersection of flat hyperplanes. When we seek to optimize a linear function

c^T x

, we are essentially sliding a 'level plane' (the objective gradient) across this diamond until it catches the furthest possible edge or face. Because the constraints

Ax = b

and

x \geq 0

define a convex polyhedron, any linear slice through it is inherently well-behaved. If we find two distinct points

x_1

and

x_2

that both yield the perfect objective value

z^*

, the line segment connecting them must stay entirely within the feasible region. Moreover, since our objective is linear, the 'cost' along this line segment cannot dip lower than

z^*

(because

z^*

is the global minimum) nor can it rise higher than

z^*

(due to the linearity of the transformation). Thus, the entire segment remains trapped at the optimal level, confirming that the optimal set is inherently convex.

CAUTION

Institutional Warning.

Students often struggle to distinguish between the 'convexity of the feasible region' and the 'convexity of the optimal set.' Remember: the feasible region is convex because linear inequalities define half-spaces, and the optimal set is a specific level-set slice of that region.

Academic Inquiries.

What happens if the objective function is strictly convex but not linear?

If the objective function were strictly convex, the optimal solution would be unique, resulting in a degenerate optimal set containing only a single point, which is trivially convex.

Does this proof rely on the assumption that an optimal solution exists?

Yes. If the feasible region is empty or the objective is unbounded, the set of optimal solutions is empty, which is vacuously a convex set.

Standardized References.

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

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 All Optimal Solutions to a Linear Program Form a Convex Set: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/linear-and-integer-programming/proof-that-all-optimal-solutions-to-a-linear-program-form-a-convex-set

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