  Many mathematical optimization algorithms fail to sufficiently explore the
solution space of high-dimensional nonlinear optimization problems due to the
curse of dimensionality. This paper proposes generative models as a complement
to optimization algorithms to improve performance in problems with high
dimensionality. To demonstrate this method, a conditional generative
adversarial network (C-GAN) is used to augment the solutions produced by a
genetic algorithm (GA) for a 311-dimensional nonconvex multi-objective
mixed-integer nonlinear optimization. The C-GAN, composed of two networks with
three fully connected hidden layers, is trained on solutions generated by the
GA, and then given sets of desired labels (i.e., objective function values),
generates complementary solutions corresponding to those labels. Six
experiments are conducted to evaluate the capabilities of the proposed method.
The generated complementary solutions are compared to the original solutions in
terms of optimality and diversity. The generative model generates solutions
with objective functions up to 100% better, and with hypervolumes up to 100%
higher, than the original solutions. These findings show that a C-GAN with even
a simple training approach and simple architecture can highly improve the
diversity and optimality of solutions found by an optimization algorithm for a
high-dimensional nonlinear optimization problem.
