For general nonlinear optimization, let the primal problem be $\min f(x)$ subject to $f_i (x) \leq 0$ for $i = 1, \ldots, m$ and $h_i(x ） = 0 $ for $i = 1, \ldots, m$. The dual problem is given by $\max g(\lambda , \nu)$ subject to $\lambda \geq 0$. Here $\lambda $ is the dual variable corresponding to the inequality constraints and $\nu$ is the dual variable for equality constraints.
Weak duality theorem states that the optimal value of primal problem ($p^*$) always lower bounds that of the dual ($d^*$), and these values can be $+/- \infty$. This means that, the following three case are also allowed:
(1) both primal and dual are infeasible; (2) primal infeasible ($p^* = +\infty$) and dual has a finite optimal value $d^*$ (3) primal has a finite optimal value ($p^*$) and dual is infeasible ($d^* = -\infty$),
Are there any concrete examples for (2) and (3). Notes: Because of the strong duality theorem (with constraint qualification conditions), for (2) and (3) to hold, the problem must be something where KKT condition is not true, or even the primal problem is nonconvex.