标题： 一种高效非凸的分阶段凸优化问题重表述 显示英文标题

标题： An efficient nonconvex reformulation of stagewise convex optimization problems

摘要： 具有分阶段结构的凸优化问题出现在多个上下文中，包括最优控制、深度神经网络的验证和同序回归。现成的求解器可以解决这些问题，但可能扩展性较差。我们开发了一种非凸重写方法，旨在利用这种分阶段结构。我们的重写方法只有简单的边界约束，可以通过投影梯度方法及其加速变体进行求解。该方法自动生成原始凸问题的一系列可行的原始和对偶解，使最优性认证变得容易。我们建立了非凸公式的理论性质，表明它几乎不存在虚假局部最小值，并且与凸问题具有相同的全局最优解。我们修改了PGD以避免虚假局部最小值，使其始终收敛到全局最小值。对于神经网络验证，我们的方法仅需几个梯度步骤即可获得较小的对偶间隙。因此，它可以比现成和专用求解器更快地解决大规模验证问题。 显示英文摘要

摘要： Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly. We develop a nonconvex reformulation designed to exploit this staged structure. Our reformulation has only simple bound constraints, enabling solution via projected gradient methods and their accelerated variants. The method automatically generates a sequence of primal and dual feasible solutions to the original convex problem, making optimality certification easy. We establish theoretical properties of the nonconvex formulation, showing that it is (almost) free of spurious local minima and has the same global optimum as the convex problem. We modify PGD to avoid spurious local minimizers so it always converges to the global minimizer. For neural network verification, our approach obtains small duality gaps in only a few gradient steps. Consequently, it can quickly solve large-scale verification problems faster than both off-the-shelf and specialized solvers.