sbx crossover always returns the lower SBX child, so crossed genes drift towards the lower bound #369

Version: PyGAD 3.7.0 (same code on master)

Found while benchmarking. crossover_type="sbx" always produces the lower of the two SBX children, so every crossed gene lands below its parents' midpoint.

pygad/utils/crossover.py L321-L331:

beta = 1.0 + 2.0 * min(y1 - lower, upper - y2) / (y2 - y1)
alpha = 2.0 - pow(beta, -(eta + 1.0))
rand_u = numpy.random.random()
if rand_u <= 1.0 / alpha:
    beta_q = pow(rand_u * alpha, 1.0 / (eta + 1.0))
else:
    beta_q = pow(1.0 / (2.0 - rand_u * alpha), 1.0 / (eta + 1.0))

child = 0.5 * ((y1 + y2) - beta_q * (y2 - y1))

Here y1 <= y2 and beta_q > 0, so child < (y1 + y2) / 2 every time.

SBX (Deb & Agrawal 1995, "Simulated binary crossover for continuous search space", Complex Systems 9) makes two children, spread symmetrically around the parents' mean:

c1 = 0.5 * [(y1 + y2) - βq (y2 - y1)]
c2 = 0.5 * [(y1 + y2) + βq (y2 - y1)]

and gives each offspring one of them at random (in the bounded version of Deb's NSGA-II code, c1 uses the spread towards the lower bound and c2 the spread towards the upper bound). Always taking c1 moves every crossed gene downwards, so the operator isn't neutral: repeated crossover drifts the population towards the lower bounds. On problems whose optimum is at the lower bound, e.g. ZDT1-3, where the optimal x2..xn are 0, this bias helps; on others it hurts.

Suggested fix: take c1 or c2 with probability 1/2.

-                child = 0.5 * ((y1 + y2) - beta_q * (y2 - y1))
+                if numpy.random.random() < 0.5:
+                    child = 0.5 * ((y1 + y2) - beta_q * (y2 - y1))
+                else:
+                    child = 0.5 * ((y1 + y2) + beta_q * (y2 - y1))
                 child = numpy.clip(child, lower, upper)

(For the exact bounded form, compute a separate beta for c2 from upper - y2, as in Deb's code; the min(...) above already keeps both children within the bounds.)