Page 148 - Data Science Algorithms in a Week

132 Alfonso T. Sarmiento and Edgar Gutierrez

Step 7) Neighborhood best updating: Determine the neighborhood best position ˆ y (k)
i
visited so far by the whole swarm by using the formula

J(y ˆ i ( k)) = min{ J(y j ( k))}, j ∈ B
i

Step 8) Global best updating: Determine the global best position g (k)visited so far by
the whole swarm by using the formula

J(g (k) ) = min{ J(y (k ))}, i = 1,..,N.
i
If J(g (k) ) < J(g (k - 1) ) then set k’ = k

Step 9) Stopping criteria: If the maximum number of iterations is achieved then stop,

g*= g (k) is the optimal solution; otherwise go to step 2.

Local Search: Powell Hill-Climbing Algorithm

PHC method basically uses one-dimensional minimization algorithms to solve multi-
dimensional optimization problems. The procedure searches into a region by constructing
a set of linearly independent, mutually “non-interfering” or conjugate search directions
and applies linear minimization to move into each direction (Press et al. 1992). The
number of conjugate directions coincides with the dimension of the search space and
their linear independence guarantees the whole search space can be covered. The use of
conjugate directions has the advantage that minimization in one direction is not interfered
by subsequent minimization along another direction, avoiding endless cycling through
the set of directions.
The steps of the algorithm are described in the following lines:

Step 1) Initialization:
 Set iteration k = 0

 Set the initial search point Z 0 = [z 1 z , 2 ,.., z n p ] as the optimal solution of the

PSO algorithm, i.e., Z = g *
0
 Initialize directions ud to the basis vectors, i.e., ud = ed, d = 1,..,np, where
e 1 = 0 , 1 [ ,.., 0 ],e 2 = 1 , 0 [ ,.., 0 ],...,e n p = 0 , 0 [ ,.., ] 1

Step 2) Define the iteration start point: Set S = Z k
0