Perhaps you can check this article:

Article Numerical methods for large-scale nonlinear optimization

I have solved many problems using Lagrange multipliers, yet never a single one by "trying every state of the variables" so I don't understand what you are asking. Perhaps the most common application is linear least squares regression with linear constraints, which yields the attached augmented matrix.

Is this linear or nonlinear optimization?

Dudley J Benton Thank you for your reply. I should correct my words. My question is particularly about dealing with inequality constraints. By different state of variables, I meant the Lagrange multipliers. As based on the constraints being active or not, the size of these matrix equations changes.

My question is a general, though our problem is nonlinear.

This type of problem (inequality constraints) is often approximated by forcing the Lagrange multipliers to be integers. Sometimes in this context the constraints are called "treatments" which isn't how I would use the word but appears in the literature and may help you search. I suspect the term "treatment" entered the literature when researchers were trying different techniques to effect chemical reactions or process medical data, so that explains the awkward terminology. You can test some smaller problems using the Solver feature in an Excel spreadsheet. You will have to enable the Solver AddIn from the Tools/AddIns menu. Set up the problem (say to minimize a residual for some data). When you add a constraint there is a drop-down menu that has =, int, bin. Thus you can force a constraint to be an integer. Then you can set it to >=0 and

What you are searching I think is here: https://www.mit.edu/~dimitrib/Constrained-Opt.pdf

Hi,

I would try using an interior-point method. They replace inequality constraints with a barrier term (a nonlinear penalization in the objective function). An alternative interpretation is the use of Newton's method on the perturbed optimality conditions (in which the complementarity equations are relaxed).

I would recommend you use Ipopt, which is a mature and robust code, and can solve very large problems. It is available here through AMPL and GAMS interfaces: https://neos-server.org/neos/solvers/index.html

It is also available through JuMP.

The code is described here: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.630.1456&rep=rep1&type=pdf

I agree with Charlie Vanaret. Interior-point method is a fast and robust method for large scale optimization problems. Use a well-experimented package such as Ipopt and you should be alright.