Question

In class we discussed the hard-margin linear support vector machine (SVM) for two-class classification which has the following primal optimization problem: Minimize 1/2||w||^2 subject to y^(?)(w^T x^(?) + w_0) ? 1, ??. We make two changes here to generalize the problem formulation: • We introduce a feature map ?(·) that corresponds to some kernel function K(·,·) such that K(x, x') = ?(x)^T ?(x'). • We change the right-hand side of the inequality constraints from 1 to some arbitrary constant ? > 0. As a result, the new primal optimization problem is as follows: Minimize 1/2||w||^2 subject to y^(?)(w^T ?(x^(?)) + w_0) ? ?, ??. (a) Give the Lagrangian function that corresponds to the primal optimization problem above by introducing appropriate Lagrange multipliers. (b) Derive the dual optimization problem that corresponds to the primal problem above. Express the dual optimization problem in terms of K(·,·) instead of ?(·). (c) Does the new optimization problem for any positive real value of ? give the same solution for the optimal separating hyperplane as the one for ? = 1? Explain your answer.

          In class we discussed the hard-margin linear support vector machine (SVM) for two-class
classification which has the following primal optimization problem:
Minimize 1/2||w||^2
subject to y^(?)(w^T x^(?) + w_0) ? 1, ??.
We make two changes here to generalize the problem formulation:
• We introduce a feature map ?(·) that corresponds to some kernel function K(·,·)
such that K(x, x') = ?(x)^T ?(x').
• We change the right-hand side of the inequality constraints from 1 to some arbitrary
constant ? > 0.
As a result, the new primal optimization problem is as follows:
Minimize 1/2||w||^2
subject to y^(?)(w^T ?(x^(?)) + w_0) ? ?, ??.
(a) Give the Lagrangian function that corresponds to the primal optimization problem
above by introducing appropriate Lagrange multipliers.
(b) Derive the dual optimization problem that corresponds to the primal problem above.
Express the dual optimization problem in terms of K(·,·) instead of ?(·).
(c) Does the new optimization problem for any positive real value of ? give the same
solution for the optimal separating hyperplane as the one for ? = 1? Explain your
answer.

In class we discussed the hard-margin linear support vector machine (SVM) for two-class
classification which has the following primal optimization problem:
Minimize 1/2||w||^2
subject to y^(?)(w^T x^(?) + w0) ? 1, ??.
We make two changes here to generalize the problem formulation:
• We introduce a feature map ?(·) that corresponds to some kernel function K(·,·)
such that K(x, x') = ?(x)^T ?(x').
• We change the right-hand side of the inequality constraints from 1 to some arbitrary
constant ? > 0.
As a result, the new primal optimization problem is as follows:
Minimize 1/2||w||^2
subject to y^(?)(w^T ?(x^(?)) + w0) ? ?, ??.
(a) Give the Lagrangian function that corresponds to the primal optimization problem
above by introducing appropriate Lagrange multipliers.
(b) Derive the dual optimization problem that corresponds to the primal problem above.
Express the dual optimization problem in terms of K(·,·) instead of ?(·).
(c) Does the new optimization problem for any positive real value of ? give the same
solution for the optimal separating hyperplane as the one for ? = 1? Explain your
answer.

Added by Angelica T.

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