# Understanding the constraint of SVM optimization problem

I'm learning SVM (support vector machines) from this book. I understand formulations of functional and geometric margins, it's also clear that we want to maximize geometric margin in order to find the optimal hyperplane, which separates the data points optimally.

What I don't understand is the constraint of the optimization problem. Following problem is given in the mentioned book:

$$\max_{w, b} M$$ $$s.t. \gamma_{i} >= M, i = 1,2...m$$

Where $$M$$ is the geometric margin of the hyperplane and $$\gamma_i$$ is geometric margin of a single data point $$(x_i, y_i)$$. Which is given by:

$$\gamma_i = y_i \left(\frac{w \cdot x_i}{||w||} + \frac{b}{||w||}\right)$$

Why we need this constraint? Couldn't we solve the problem without this constraint? I don't find an intuitive reason behind the constraint.

• If you find any hyperplane where each point is correctly classified, then it is still a solution to SVM, yet the solution itself is not generalized enough. Note that the constraint itself states that the distance between any point and the separating hyperplane is at least the margin thus, it's correctly classified since 1. because of the constraint on the distance, and 2. since constraint holds only when the point is correctly classified (otherwise u will have $\gamma_i \geq M$ but $\gamma_i < 0$). Note that $M > 0$. Mar 24 '19 at 12:11