Dual optimization problem has no stationary point

I want to solve an optimization problem $\max_x f(x)$ subject to equality constraint $c(x) = 0$ with Lagrange multipliers method $$\min_\lambda L(\lambda) = \min_\lambda [ \max_x (f(x) + \lambda c(x)) ]$$

I can relatively easy evaluate dual function $L(\lambda)$, however, it's not defined for all $\lambda$ since equation $\frac{d}{dx}(f(x) + \lambda c(x)) = 0$ may not have real root (I'm looking for real solution).

Does it mean that my original constrained problem has no solution? Are the there any walkarounds one may suggest?

Function f(x) is not linear, but could be assumed convex if necessary.

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If the function $f$ is convex and differentiable, $c$ is continious differentiable. If $E = \{x | x \in Def(f) and c(x) = 0 \}$ doesn't have a solution then your constrained problem don't have any solution otherwise you should find a solution using the lagrangian. I might help you if your put the explicit problem –  Samatix Jul 25 '13 at 12:38