The idea of shrinking is to remove some variables whose values have been equal
to the bounds 0 or C for a long time, and that will probably
not change anymore.
To do this, we use the fact that
the problem (3) under the constraints
(4) and (5) if and only if
there exists numbers
Note that if , then the corresponding variable is equal to 0. Also, if , then the corresponding variable is equal to C. The idea is thus to search at each iteration for variables , and that verify as well as possible3 the equations (24)-(28), and to remove a variable whose value is equal to 0 if its coefficient is strictly positive (or just above a constant ) during a given number of iterations. Using the same idea, we also eliminate a variable whose value is equal to Cif its coefficient stays strictly positive for a sufficient number of iterations.
we start by estimating
(note that if
Then if we have