I am confused about the general rule of Lagrangian multiplier. (which usully use for SVM). I could not find a good book or paper that explain it completely and clearly.

Suppose we have

$$minimize: w^{2}$$

Subject to $$f(x) \leq 0$$

Lagragian multiplier say we can solve the following

$$min : \; \; w^{2}+\lambda f(x) $$

subject to $$ \lambda \geq 0$$

Then we solve gradian equations and substitute back in Lagrangian.

1- how we make sure the solution satisfy f(x) <0 ?

2- what is extra constraints in dual representation?



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