Convexity of the Feasible Region in Linear Programming

Q: Why is the intersection of half-spaces always convex?

A half-space $ \{\mathbf{x} : \mathbf{a}^T \mathbf{x} \leq b \} $ is convex. Since the intersection of any collection of convex sets remains convex, the feasible region, formed by intersecting multiple half-spaces, inherits this property.

Analytical Intuition.

Imagine the feasible region as a vast, crystalline chamber constructed from flat, obsidian walls of linear constraints. When we speak of 'convexity', we are describing the structural integrity of this chamber. If you were to pick any two points

\mathbf{x}_1

and

\mathbf{x}_2

residing anywhere inside this room—whether on the floor, clinging to the ceiling, or floating in the center—the straight-line path connecting them never pierces a wall to exit into the forbidden exterior. Mathematically, the linear nature of our constraints

A\mathbf{x} \leq \mathbf{b}

ensures that the weighted average

\lambda \mathbf{x}_1 + (1 - \lambda) \mathbf{x}_2

maintains the balance of the inequality. Because each constraint is a half-space, and the intersection of any number of half-spaces is inherently convex, our feasible region forms a 'convex polytope'. This is the 'Golden Rule' of optimization: because the terrain has no holes, no hidden coves, and no jagged inward dents, any local peak we find on this landscape is guaranteed by necessity to be the global summit.

Institutional Warning.

Students often conflate 'convexity' with 'compactness'. A feasible region can be convex but unbounded (e.g., in a minimization problem with an open constraint set). Convexity guarantees that no 'local' traps exist, but it does not guarantee that a finite optimal solution exists without further bounds.

Academic Inquiries.

Why is the intersection of half-spaces always convex?

A half-space $\{\mathbf{x} : \mathbf{a}^T \mathbf{x} \leq b \}$ is convex. Since the intersection of any collection of convex sets remains convex, the feasible region, formed by intersecting multiple half-spaces, inherits this property.

Does non-convexity make optimization impossible?

Not impossible, but significantly harder. In non-convex optimization, gradient-based methods can get trapped in local optima, requiring global search heuristics or stochastic methods to locate the true global maximum.

NICEFA Visual Mathematics. (2026). Convexity of the Feasible Region in Linear Programming: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/fundamentals-of-optimization/convexity-of-the-feasible-region-in-linear-programming

Visualizing...

The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

Why is the intersection of half-spaces always convex?

Does non-convexity make optimization impossible?

Standardized References.

Weierstrass Extreme Value Theorem: Guaranteeing Existence of Optima

Local Optima are Global Optima for Convex Functions

Hessian Matrix and Second-Order Optimality Conditions

Jensen's Inequality for Convex Functions

Institutional Citation

Dominate the Logic.

Visualizing...

The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

Why is the intersection of half-spaces always convex?

Does non-convexity make optimization impossible?

Standardized References.

Related Proofs Cluster.

Weierstrass Extreme Value Theorem: Guaranteeing Existence of Optima

Local Optima are Global Optima for Convex Functions

Hessian Matrix and Second-Order Optimality Conditions

Jensen's Inequality for Convex Functions

Institutional Citation

Dominate the Logic.