Optimization of the Parameters of a Piecewise Linear Decision Rule

Abstract

This chapter is devoted to the solution of the problem formulated in Section 2.7—the optimal choice of the parameters of a piecewise linear decision rule with a bounded number of linear functions. We recall that a decision rule of the form

$$ d = \arg \mathop {\max }\limits_k {\mathop f\limits^ \sim _k}\left( \upsilon \right) $$

is said to be piecewise linear if the discriminant functions ${\tilde f_k}(\upsilon )$ are. This means that, depending on the value of v, the value of ${{\tilde f}_k}(\upsilon )$ coincides with the value of one of the given linear forms in the components ${\upsilon _i}$. The number of these forms is finite. For instance, the function

$$ f(\upsilon ) = \mathop {\max }\limits_\zeta \left( {{c_0}(\zeta ) + \sum\limits_{i = 1}^N {{\upsilon _i}{c_i}(\zeta )} } \right) $$

is piecewise linear.