  Topology optimization problems often support multiple local minima due to a
lack of convexity. Typically, gradient-based techniques combined with
continuation in model parameters are used to promote convergence to more
optimal solutions; however, these methods can fail even in the simplest cases.
In this paper, we present an algorithm to perform a systematic exploratory
search for the solutions of the optimization problem via second-order methods
without a good initial guess. The algorithm combines the techniques of
deflation, barrier methods and primal-dual active set solvers in a novel way.
We demonstrate this approach on several numerical examples, observe
mesh-independence in certain cases and show that multiple distinct local minima
can be recovered.
