Template polyhedra and bilinear optimization

In this paper, we study the template polyhedral abstract domain using connections to bilinear optimization techniques. The connections between abstract interpretation and convex optimization approaches have been studied for nearly a decade now. Specifically, data flow constraints for numerical domains such as polyhedra can be expressed in terms of bilinear constraints. Algorithms such as policy and strategy iteration have been proposed for the special case of bilinear constraints that arise from template polyhedra wherein the desired invariants conform to a fixed template form. In particular, policy iteration improves upon a known post-fixed point by alternating between solving for an improved post-fixed point against finding certificates that are used to prove the new fixed point. In the first part of this paper, we propose a policy iteration scheme that changes the template on the fly in order to prove a target reachability property of interest. We show how the change to the template naturally fits inside a policy iteration scheme, and thus, propose a scheme that updates the template matrices associated with each program location. We demonstrate that the approach is effective over a set of benchmark instances, wherein, starting from a simple predefined choice of templates, the approach is able to infer appropriate template directions to prove a property of interest. However, it is well known that policy iteration can end up “stuck” in a saddle point from which future iterations cannot make progress. In the second part of this paper, we study this problem further by empirically comparing policy iteration with a variety of other approaches for bilinear programming. These approaches adapt well-known algorithms to the special case of bilinear programs as well as using off-the-shelf tools for nonlinear programming. Our initial experience suggests that policy iteration seems to be the most advantageous approach for problems arising from abstract interpretation, despite the potential problems of getting stuck at a saddle point.