Wolfe Conditions and capturing the minimizer

Hi guys I am learning about line search algorithms. In general the idea is that we want to solve the minimization problem

$$\min_\alpha f(x_k+ \alpha p_k)$$

Then we use alpha as our step-size. In practice this is expensive to solve so we get a close enough alpha not nessesary the best. One of the most used conditions to ensure we do decrease the fuchtion are the Wolfe conditions

$$f(\mathbf{x}_k+\alpha_k\mathbf{p}_k)\leq f(\mathbf{x}_k)+c_1\alpha_k\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k)$$ and $$-\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k+\alpha_k\mathbf{p}_k) \leq -c_2\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k)$$

My question is are we guaranteed that among the alphas that the Wolfe conditions we have the actual solution to the minimization problem? To me there is not an obvious answer.