Secondary Structure Classification of Isoform Protein Markers in Oncology - Mathematical Approaches to Polymer Sequence Analysis and Related Problems - page 53

each feature vector must be nearer to the assigned barycenter of its own class than
to any other barycenter. Should the barycenter be null, this is immediately verified,
while if it is non zero, this must be imposed. The inequality ( 3.8 ) indicates that the
vectors z 2 R

nmc are binary.
The solution will determine that each pattern of the training set is nearer to a
barycenter of its own class than to a barycenter of another class. Each barycenter
has the class label of the patterns assigned to it, which will belong by construction
to a single class. This defines a partition of the pattern space.
A new pattern can be assigned to a class by determining its distance from each
barycenter formed by the algorithm and then assigning the pattern to the class of the
barycenter to which it is nearest.
In addition, the optimization problem ( 3.1 )-( 3.8 ) may be formulated as a nonlin-
ear complementarity problem facilitating the proof of convergence and termination
of the algorithm. The nonlinear complementarity formulation is a statement of the
Karush-Kuhn-Tucker condition of an optimization problem [ 19 ], and therefore, one
of the solutions of the nonlinear complementarity problem will be a solution to that
optimization problem
To demonstrate that the algorithm will converge to an optimal solution, consider
the domain of the optimization problem to be over

mc

and y 2 R

R N a convex space.

F W R N ! R N

F. w / 0 (3.9)

w 2 R N

w 0 (3.10)

w T F. w / D 0;

(3.11)

where w comprises all the variables to be determined, the binary variables and the
lagrangian multipliers of the inequalities.
This problem can be written as a variational inequality:

F. w / T . u w / 0 (3.12)

w 0 (3.13)

8 u 0: (3.14)

The solutions of the two problems are identical and the following results have been
demonstrated [ 5 ].

Theorem 3.3. Let K R N be a non empty, convex and compact set and let F W
K ! K be a continuous mapping. The following are equivalent:
1. There exists a fixed point w 2 K for this mapping,
2. The variational inequality ( 3.12 ) and ( 3.14 ) has a solution,
3. The nonlinear complementarity problem ( 3.9 )-( 3.11 ) has a solution

Consider the nonlinear complementarity problem ( 3.9 )-( 3.11 ) and limit its solution
to occur within a trust region set, defined by a set of linear inequalities which can
be so indicated:

D w d; (3.15)