Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

The above theorem is called the fundamental theorem of GA or the
Schema theorem. It is evident from the Schema theorem that for a given set
of values of d(H), O(H), L, pc and pm , the population of schema H at the
subsequent generations increases exponentially when fH > fav. This, in fact,
directly follows from the difference equation:
m H ( t +1) – m H (t) ≥ ( fH / fav –1 / K ) K mH (t) (15.9)
where K = 1 – pc d (H) / (L – 1) –pm O(H). (15.10)
⇒ ∆mH (t) ≥ K ( f H / fav -1/ K ) m H (t). (15.11)
Replacing ∆ by (E –1), where E is the extended difference operator, we find
(E –1- K1) m H (t) ≥ 0 (15.12)
where K1 = K ( f H / fav -1 / K). (15.13)
Since m H (t) in equation (15.12) is positive, E ≥ (1 +K1 ). Thus, the
solution of (15.12) is given by
m H (t) ≥ A ( 1 + K1) t (15.14)
where A is a constant. Setting the boundary condition at t = 0, and substituting
the value of K1 by (14.13) therein, we finally have:
m H (t) ≥ mH (0) ( K fH / fav) t (15.15)
Since K is a positive number, and f H / fav > 1, mH (t) grows
exponentially with iterations. The process of exponential increase of mH (t)
continues until some iteration r, when fH approaches fav. This is all about the
proof of the schema theorem.
15.4 The Markov Model for
Convergence Analysis
To study the convergence of the GA, let us consider an exhaustive set of
population states, where ‘state’ means possible members (chromosomes) that
evolve at any GA cycle. As an illustration, let us consider 2-bit chromosomes