The SVM dual objective

Score a set of dual variables and check they are legal

The soft-margin SVM can be solved in its dual form, where one variable alpha_i per training point carries the whole solution. The dual objective to maximize is W(alpha) = sum(alpha) - 0.5 * sum_ij alpha_i alpha_j y_i y_j K_ij, with labels y_i in {-1, +1} and kernel matrix K.

Implement two functions.

• dual_objective(alpha, y, K): returns W(alpha) as a float. alpha and y have length n and K is an n by n kernel matrix.
• is_dual_feasible(alpha, y, C, tol=1e-9): returns True when every alpha_i lies in [0, C] (within tol) and sum(alpha_i * y_i) is zero (within tol). Otherwise it returns False.

Constraints

• alpha and y must have length n, and K must be n by n. Otherwise raise ValueError.
• C < 0 raises ValueError.
• Do not modify the inputs.

Hints