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.

中文翻译：

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模板多面体和双线性优化

在本文中，我们使用与双线性优化技术的连接来研究模板多面体抽象域。抽象解释和凸优化方法之间的联系已经研究了近十年。具体来说，诸如多面体之类的数值域的数据流约束可以用双线性约束来表示。已经针对模板多面体产生的双线性约束的特殊情况提出了诸如策略和策略迭代之类的算法，其中所需的不变量符合固定的模板形式。特别是，策略迭代通过在求解改进的后固定点与查找用于证明新固定点的证书之间交替来改进已知的后固定点。在本文的第一部分，我们提出了一种策略迭代方案，可以动态更改模板，以证明感兴趣的目标可达性属性。我们展示了对模板的更改如何自然地适应策略迭代方案，因此，提出了一种更新与每个程序位置相关联的模板矩阵的方案。我们证明该方法在一组基准实例上是有效的，其中，从简单的预定义模板选择开始，该方法能够推断适当的模板方向以证明感兴趣的属性。然而，众所周知，策略迭代最终可能会“卡在”一个鞍点，未来的迭代无法从中取得进展。在本文的第二部分，我们通过经验比较策略迭代与各种其他双线性规划方法来进一步研究这个问题。这些方法使众所周知的算法适用于双线性程序的特殊情况，并使用现成的工具进行非线性编程。我们最初的经验表明，尽管存在陷入鞍点的潜在问题，但策略迭代似乎是解决由抽象解释引起的问题的最有利方法。