$$\max \ \min[\alpha x_1, \beta x_2, \gamma x_3] \ \ \text{s.t.} \ \lambda_1 x_1 + \lambda_2x_2 + \lambda_3x_3 = c, \\\ \alpha, \beta, \gamma, \lambda_i, c \ \text{are constants}$$
Well, that function is not differentiable , so what methods can be applied to solve for for the optimal values of $x_1, x_2$ and $x_3$? Is knowledge of the $\lambda's$ and $c$ necessary, to at least some degree, or does a general approach/solution exist?
You can reformulate it to be a Linear Program:
\begin{align} \max &\quad z\\ z &\leq \alpha x_1 \\ z &\leq \beta x_2 \\ z &\leq \gamma x_3 \\ \lambda_1 x_1 +\lambda_2 x_2 +\lambda_3 x_3 &= c \end{align}
which you can now feed into a Linear Programming solver and get a answer very easily. A closed-form expression for the optimum probably exists and this can be explored if you write the dual of this problem and use the sign constraints on the variables and parameters.
When the objective function is not differentiable, a common procedure is the use of subgradient method. Subgradient Methods - Stanford.
