On pp. 248 of Boyd and Vandenberghe: suppose 1) strong duality holds, 2) the dual optimal is attained at $(\lambda^*, \nu^*)$, 3) the dual function $L(x, \lambda^*, \nu^*)$ has the unique minimizer $x^*$, 4) $x^*$ is primal feasible.
How do those assumptions imply the primal optimum is also attained and $x^*$ is in fact the primal optimal solution?