Lagrange multiplier is negative in SVM

Question

P.S: I am beginner also I am not asking help in exam problem I have that problem in exam: Suppose you have 10 data points :(8,6),(9,2),(7,4),(2,0) ∈ x C1 (+ve) and (0,2),(-2,0),(3,5),(1,3.5) ∈ C2(-ve):

1. plot these training points and construct by inspection the weight vector for the optimal hyperplane ,and the optimal margin

2. what are the support vectors\

3. Construct the solution by finding the Lagrange multiplier

my answer: 1,2- svm-margin graph

3- After solve this optimization problem: min (1/2)w(transpose)w s.t : 2w1+w0 >=1, 6w1+4w2+w0>=1, 2*w2+w0<=-1.

I get one of the Lagrange multipliers is negative is it possible? if yes ,why? or it is just miscalculation from me.

Answer 1

The Lagrange multipliers $\mathbf{a}$ are found by solving the constrained minimization problem $$ \phi(\mathbf{a})= -\mathbf{1}_N^T \mathbf{a} +\frac{1}{2} \mathbf{a}^T \mathbf{H} \mathbf{a}, s.t. \mathbf{a} \ge \mathbf{0} $$ where $H_{ij}= y_i y_j \mathbf{x}_i^T \mathbf{x}_j$

The multipliers are either null (for non supporting vectors) or strictly positive otherwise.