Need help with Lagrange Multipliers in Support Vector Machine

Question

In linear support vector machine with binary classification if we know all the support vectors, geometrically we can create hyperplanes on both sides of the main hyperplane using these support vectors dimension, and main hyperplane is at the middle of these hyperplane normal to the line connecting these hyperplanes, so why do we need Lagrange multiplier to calculate the equation of main hyperplane? Equation of hyperplane is assumed to be W.X+B=0 and we get value of W=ΣᾁYX

Answer 1

if we know all the support vectors, geometrically we can create hyperplanes on both sides

Strictly speaking, support vectors do not necessarily determine hyperplanes uniquely. For example, if you only have two data points (one in each class), then the support vectors are obvious, but the hyperplane still needs to be found. But this is a minor point. The real answer is: you need Lagrange multipliers (or some form of convex minimization) because you don't know yet what support vectors are going to be.