Why do the Gauss‑Markov assumptions guarantee OLS is the best linear unbiased estimator?

What Is a Basic Feasible Solution?

In standard form, an LP has:

• n decision variables
• m constraints

A BFS is constructed by:

• choosing m variables to serve as basic variables,
• setting the other n − m variables to zero,
• solving the resulting m × m system.

If the solution is feasible, it is a basic feasible solution.

Why Basic Feasible Solutions Are Corner Points

1. Corner points occur where m constraints bind

A vertex of the feasible region is formed where m constraints intersect. These constraints form a system of m equations in m unknowns.

2. A BFS solves exactly m equations

A BFS is obtained by selecting m basic variables and solving m equations — the same structure that defines a vertex.

3. Algebraic and geometric definitions match

The algebraic definition of a BFS corresponds exactly to the geometric definition of a corner point.

4. Every BFS is a corner point of the feasible polyhedron

If the BFS is feasible, it lies at a vertex of the feasible region. This is why simplex moves from BFS to BFS — it is moving from corner to corner.

5. The Fundamental Theorem of Linear Programming

If an LP has an optimal solution, at least one optimal solution is a BFS (i.e., a corner point).

How to Identify Basic Feasible Solutions (Step by Step)

Step 1: Convert the LP to standard form

Write all constraints as equations by adding slack or surplus variables. Ensure all variables satisfy nonnegativity.

Step 2: Count variables and constraints

Let:

• n = number of variables
• m = number of constraints

A BFS uses exactly m basic variables.

Step 3: Choose m variables to be basic

Set the remaining n − m variables to zero. This reduces the system to m equations in m unknowns.

Step 4: Solve the reduced system

Solve for the m basic variables. If all basic variables are ≥ 0, the solution is feasible.

Step 5: Check feasibility

A BFS must satisfy:

• all equality constraints,
• all ≤ constraints,
• all nonnegativity constraints.

Step 6: Interpret the BFS geometrically

The BFS corresponds to the intersection of the m binding constraints — a corner point of the feasible region.

Step 7: Connect BFS to simplex movement

Simplex moves from one BFS to another by pivoting, replacing one basic variable with one nonbasic variable and moving to an adjacent corner.

Numerical Example

Consider the LP:

\[ \text{Max } z = 3x_1 + 2x_2 \]

\[ \begin{aligned} x_1 + x_2 &\le 6 \\ x_1 &\le 4 \\ x_2 &\le 5 \\ x_1, x_2 &\ge 0 \end{aligned} \]

Corner points (each is a BFS)

• (0,0)
• (4,0)
• (4,2)
• (1,5)
• (0,5)

Each point is obtained by selecting m = 2 binding constraints and solving the resulting system.

Optimal BFS

Evaluating the objective shows the maximum occurs at (4,2), which is both a BFS and a corner point.