Optimization problem with convex quadratic objectives and linear constraints

Can be solved using QP.

Lagrange duality to get the optimization problem's dual form,

- Allow us to use kernels to get optimal margin classifiers to work efficiently in very high dimensional spaces.
- Allow us to derive an efficient algorithm for solving the above optimization problem that will typically do much better than generic QP software.

**Lagrangian Duality in brief**

The Primal Problem

The generalized Lagrangian :

the 𝛼's (𝛼≥0) and β' are called the Lagrange multipliers Lemma:

A re-written Primal:

The Primal Problem

The Dual Problem :

Theorem (weak duality) :

Theorem (strong duality):

If there exist a saddle point of L(w,𝛼,β), we have

**The KKT conditions**

If there exists some saddle point of L, then it satisfies the following "Karush-Kuhn-Tucker" (KKT) conditions:

**Theorem**: If w* , a* and b* satisfy the KKT condition , then it is also a solution to the primal and the dual problems.

**Support Vectors**

- Only a few 𝛼i's can be nonzero
- Call the training data points whose 𝛼i's are nonzero the support vectors

**Solving the Optimization Problem**

Quadratic Programming with linear constraints

Lagrangian Function ↕️

**The Dual Problem**

Now we have the following dual opt problem:

This is a quadratic programming problem.

- A global maximum of 𝛼i can always be found.

**Support Vector Machine**

Once we have the Lagrange multipliers {𝛼i} we can reconstruct the parameter vector w as a weighted combination of the training examples:

For testing with a new data z

- Compute

Note: w need not be formed explicitly

**The Solving the Optimization Problem**

The discriminant function is:

It relies on a dot product between the test point **x**and the support vector xi.

Solving the optimization problem involved computing the dot products xiTxj between all pairs of training points.

The optimal w is a linear combination of a small number of data points.

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