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A proof from An Introduction to Optimization By Edwin Chong and Zak

Theorem 15.1 Fundamental Theorem of LPP. Consider a linear program in standard form.

  1. If there exists a feasible solution, then there exists a basic feasible solution;
  2. If there exists an optimal feasible solution, then there exists an optimal basic feasible solution.

    Proof. We first prove part 1. Suppose that $x = [x_1,..., x_n]^T$ is a feasible solution and it has $p$ positive components. Without loss of generality, we can assume that the first $p$ components are positive, whereas the remaining components are zero. Then, in terms of the columns of $A = [a_1,..., a_p, . . . , a_n]$ this solution satisfies

    $x_1a_1+...x_pa_p=b$.

There are now two cases to consider.

Case 1: If $a_1, a_2,..., a_p$ are linearly independent, then $p \leq m$. If $p = m$, then the solution $x$ is basic and the proof is completed. If, on the other hand, $p < m$, then, since $rank A = m$, we can find $m—p$ columns of A from the remaining $n — p$ columns so that the resulting set of m columns forms a basis. Hence, the solution $x$ is a (degenerate) basic feasible solution corresponding to the above basis.

Case 2: Assume that $a_1, a_2,..., a_p$ are linearly dependent. Then, there exist numbers $y_i, i = 1, . . . , p $ not all zero, such that $y_1a_1+..y_pa_p=0$

We can assume that there exists at least one $y_i$ that is positive, for if all the $y_i$ are nonpositive, we can multiply the above equation by $-1$. Multiply the above equation by a scalar $\epsilon$ and subtract the resulting equation from $x_1a_1+...x_pa_p=b$ to obtain $(x_1-\epsilon y_1)a_1+....+(x_p-\epsilon y_p)a_p=b$

Let $y=[y_1,..y_p,0,..0]^T$.then Then, for any $\epsilon$

we can write $A[x-\epsilon y]=b$

Let $\epsilon$ = min{$xi/yi : i = 1,..., p, yi > 0$}. Then, the first $p$ components o f $x — \epsilon y$ are nonnegative, and at least one of these components is zero. We then have a feasible solution with at most $p — 1$ positive components. We can repeat this process until we get linearly independent columns of A, after which we are back to Case 1. Therefore, part 1 is proved.

My question is why first $p$ components of $x-\epsilon y$ is non-negative.I thought to understand this by contradiction that suppose for any $i$ $i$ $\in$ {$1,2,..p$},

$x_i-\epsilon y_i<0$

$\implies x_i<\epsilon y_i$

$\implies \frac{x_i}{y_i}<\epsilon$.I dont know how to go further.I could have achieved contradiction if $\epsilon$ = min{$xi/yi : i = 1,..., p$},but they have defined $\epsilon$ only for $y_i>0$$

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We have $x\ge0$. Then $\epsilon\ge0$. If $y_i\le0$ then $x_i -\epsilon y_i\ge0$. If $y_i>0$ then $\epsilon\le x_i / y_i$, and $$ x_i - \epsilon y_i \ge x_i - \frac{x_i}{y_i} y_i=0. $$

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