Constrained and unconstrained optimization

I have always been confused about whether the equation is constrained or uncontsrained, and how to solve it. Below are the few queries

  1. How to convert a uncontsrained optimization problem to a constrained optimization problem?
    What are the techniques available?
    Like linear regression with square loss can be solved using stochastic gradient descent but if the regulariser term is added then whether this equation is constrained now or not? Because “sgd solves unconstrained optimization problem whereas Lagrange multipliers is used to solve constrained optimization problem” Is this statement true?

  2. When to use Lagrange and when to use sgd?

  3. How to determine whether the equation is constrained or not?

  4. Does langrange convert the constrained optimization problem to uncontsrained optimization problem?

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