Does KKT conditions and Lagrange Dual are really feasible when the no of variables are constraints are more?

Question

I want to find the maximum of \begin{align} \quad & f(x) = x_1x_2x_3 \\ \text{subject to} \quad & x_1+x_2+x_3=0 \\ & -1 \leq x_i \leq 1 \hspace{1ex} i=1,2,3 \\ \end{align} Now the Lagrange Dual will be \begin{equation} L(x_i, \lambda_i, \mu_i, \gamma)=-x_1x_2x_3+ \sum_{i=1}^3 \lambda_i (x_i-1) - \sum_{i=1}^3 \mu_i (x_i+1)+ \gamma \sum_{i=1}^3 x_i \end{equation} Now the KKT conditions are \begin{equation} \begin{split} \frac{\partial L}{\partial x_i} & = 0 \\ \sum_{i=1}^3 x_i &= 0 \\ \lambda_i (x_i-1)& = 0 \\ \mu_i (x_i+1)& = 0 \\ \lambda_i & \geq 0 \\ \mu_i & \geq 0 \end{split}\label{a}\tag{1} \end{equation} I feel that there are lot of conditions and that the complementary slackness conditions are not giving enough information. Since $\lambda_i, \mu_i \geq 0$, we've to check for the values manually and then eliminate the cases which doesn't bound by constraints. I feel that this procedure is tedious. If it is actually for such a simple problem, How is it being used in the real world? Do we actually verify all the cases and eliminate the ones which doesn't satisfy the conditions? Please also suggest any math tools/software that is used for constraint optimisation problems.

Answer 1

No, in practice we don't directly solve the KKT conditions as a nonlinear system of equations. Rather, in primal-dual methods, we use an iterative algorithm to produce a sequence of primal and dual solutions and check the KKT conditions at each iteration. When the KKT conditions are satisfied to within some accuracy tolerance, we stop the algorithm.

There are many software libraries for nonlinear optimization as well as high level modeling languages that make it easy to express an optimization problem and send it to a solver. It's impossible to recommend a particular package to you without knowing more about your specific problems.