  Optimization is an essential component for solving problems in wide-ranging
fields. Ideally, the objective function should be designed such that the
solution is unique and the optimization problem can be solved stably. However,
the objective function used in a practical application is usually non-convex,
and sometimes it even has an infinite set of solutions. To address this issue,
we propose to learn the solution manifold in optimization. We train a model
conditioned on the latent variable such that the model represents an infinite
set of solutions. In our framework, we reduce this problem to density
estimation by using importance sampling, and the latent representation of the
solutions is learned by maximizing the variational lower bound. We apply the
proposed algorithm to motion-planning problems, which involve the optimization
of high-dimensional parameters. The experimental results indicate that the
solution manifold can be learned with the proposed algorithm, and the trained
model represents an infinite set of homotopic solutions for motion-planning
problems.
