Add OmniOptimizer algorithm

examples/algorithms/moo/omni.py

@@ -0,0 +1,45 @@
+import matplotlib.pyplot as plt
+
+from pymoo.algorithms.moo.omni import OmniOptimizer
+from pymoo.optimize import minimize
+from pymoo.problems.multi.omnitest import OmniTest
+from pymoo.visualization.scatter import Scatter
+
+# The omni-test problem has 3^n_var equivalent Pareto subsets in the decision space that
+# all map to the same Pareto front. The key feature of the Omni-Optimizer is the crowding
+# distance defined in the variable space, which allows it to find and maintain all of those
+# subsets. Disabling it (var_crowding=False) niches only in objective space and essentially
+# recovers the NSGA-II behavior - the equivalent subsets then collapse onto a few of them.
+problem = OmniTest(n_var=2)
+
+with_var = minimize(problem, OmniOptimizer(pop_size=100),
+                    ('n_gen', 200), seed=1, verbose=False)
+
+without_var = minimize(problem, OmniOptimizer(pop_size=100, var_crowding=False),
+                       ('n_gen', 200), seed=1, verbose=False)
+
+PS = problem.pareto_set(5000)
+PF = problem.pareto_front(5000)
+
+# decision space - with variable-space niching all equivalent subsets are covered
+plot = Scatter(title="Decision Space (with variable-space niching)", labels="x")
+plot.add(PS, s=10, color="red", label="Pareto set")
+plot.add(with_var.X, s=20, color="blue", label="Obtained solutions")
+plot.do()
+plt.legend()
+
+# decision space - without it, only a subset of the equivalent regions survives
+plot = Scatter(title="Decision Space (without variable-space niching)", labels="x")
+plot.add(PS, s=10, color="red", label="Pareto set")
+plot.add(without_var.X, s=20, color="blue", label="Obtained solutions")
+plot.do()
+plt.legend()
+
+# both converge to the same Pareto front in objective space
+plot = Scatter(title="Objective Space")
+plot.add(PF, s=10, color="red", label="Pareto front")
+plot.add(with_var.F, s=20, color="blue", label="Obtained solutions")
+plot.do()
+plt.legend()
+
+plt.show()