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By Francesco Parrella

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So it often happens that a sample exit from its set and does no more verify KKT conditions. To avoid this problem, there has been ideated a way to correct these samples and move back to its original sets. The basic idea is to use the same principles applied in normal algorithm and extend them with new possible moves permitted only when a sample exit from its set. In the next paragraphs are listed all the possible variations. 1 Invalid Support Samples A support sample becomes invalid when its |θi | becomes > C.

C . 40) In this way, ∆θi and ∆b values can be updated by computing β, and ∆(xi ) values can be updated by computing γ. 7 Empty SupportSet In the special case that SupportSet is empty, values should be updated differently. 8. 8 39 Samples Moving Up until now, we defined a way to update all three sets values of ∆θc maintaining the KKT conditions consistent. Now, the problem is to identify the right value of ∆θc in all possible situations that can happen. A simple representation of support set, error set and remaining set in relation to h(x) as follows: In this image, each circle corresponds to a sample.

H(xnln ) ∆Qnln c 1 Qnln s1 · · · Qnln sls Now, γ can be defined as:     γ=   0 1 ··· 1  ∆Qn1 c   1 Qn1 s1 · · · Qn1 sls ..   + .  .. ...   ... .  ∆Qnln c 1 Qnln s1 · · · Qnln sls And rewriting the formula with new notations:   ∆h(xn1 )   ..     = γ∆θc . 40) In this way, ∆θi and ∆b values can be updated by computing β, and ∆(xi ) values can be updated by computing γ. 7 Empty SupportSet In the special case that SupportSet is empty, values should be updated differently.

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