We know we can solve an LP directly using KKT matrix method, even for QPs this works, for an example problem
$$ \min_{x_1,x_2} x_1^2 + x_2^2 \quad \textrm{s.t.} $$ $$ x_1 + x_2 = 5 $$
KKT matrix is
$$ \left[\begin{array}{ccc} Q & A^T \\ A & 0 \end{array}\right] \left[\begin{array}{c} x \\ u \end{array}\right] = \left[\begin{array}{r} -c \\ b \end{array}\right] $$
then we can solve
$$ \left[\begin{array}{ccc} 2 & 0 & 1 \\ 0 & 2 & 1 \\ 1 & 1 & 0 \end{array}\right] \cdot \left[\begin{array}{c} x_1 \\ x_2 \\ u \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 5 \end{array}\right] $$
But I usually see in lectures on LPs primal-dual methods are used, for problems such as
$$ \min_x f(x) \quad \textrm{s.t.} $$ $$ Ax = b $$ $$ h_i(x) \le 0, \quad i=1,..,m $$
Even if there were inequalities we know we can turn them into the form above. Then my question is why even bother with the primal-dual method when there is a method that works through a single linear algebra solve call? Is it done to lay some ground work for future cases when problems might not be convex, or differentiable etc?
Thanks,
For which you need to give an example of the problem.
Your problem we can solve by C-S: $$x_1^2+x_2^2=\frac{1}{2}(1^2+1^2)(x_1^2+x_2^2)\geq\frac{1}{2}(x_1+x_2)^2=\frac{25}{2}.$$ The equality occurs for $$(x_1,x_2)||(1,1),$$ which gives $x_1=x_2=\frac{5}{2},$ which says that we got a minimal value.