二分类问题下支持向量机最大间隔目标函数和约束不等式的推导

Sure, I’d be happy to answer your question about the derivation of the objective function and inequality constraints for Support Vector Machines in the case of binary classification.

The goal of a Support Vector Machine is to find the hyperplane that best separates the two classes of data. This hyperplane can be represented by a vector w and a scalar b such that:

w.T * x + b > 0 if x belongs to class 1
w.T * x + b < 0 if x belongs to class 2

We can simplify this by defining the actual class labels as y, where y = -1 for class 1 and y = 1 for class 2. Then, we can rewrite the above equations as:

y(w.T * x + b) > 0

The distance between the hyperplane and the closest data point from either class is called the margin. The goal of SVM is to maximize this margin. The margin is given by:

margin = (w / ||w||) . (x_2 - x_1)
where x_1 and x_2 are the closest data points from either class and ||w|| is the length of the vector w. Maximizing this margin is equivalent to minimizing ||w|| subject to the conditions that every data point be classified correctly. This leads to the following optimization problem:

argmin(w,b) ||w||^2
subject to: y_i(w.T * x_i + b) >= 1 for all i

This is the objective function with inequality constraints for a Support Vector Machine.

I hope this helps answer your question. Let me know if you have any more!