Consider a linear programming (LP) with inequality constraints:
$$ \text{min}\quad f(x) = c^Tx \\ \text{s.t.} \mu: Ax \leq b, \\ \quad x\in X. $$
where $X$ is a convex set. $\mu$ is the dual variable, which can be updated by
$$ \mu(k+1)=[\mu(k)+\alpha(k)(A\overline{x}(k+1)-b)]_+, \quad (1) $$
where $\alpha(k)$ is the non-summable but square-summable step size, $[]_+$ is the non-negative projection, $\overline{x}(k+1)$ is the optimal solution of
$$ \text{min}_{x\in X}\quad c^Tx+\mu(k)^TA*x, \quad (2) $$
Can this iterative method solve the problem? Assume the primal PL has a unique optimal solution. Obviously because it is linear, the optimal solution will be found among the vertices of the feasible region $S=S_1\cap S_2$, where $S_1=\{x|Ax \leq b\}, S_2=\{x\in X\}$. But the optimal solution of (2) is only found on a vertex of $S_2$, since whatever value $\mu$ takes, (2) is also a linear problem. Now, it seems that if the optimal solution of the primal PL is located at a vertex of $S_1$, the iterative method will not find it.
Is this true?