We study the geometry of convex optimization problems given in a Domain-Driven form and categorize possible statuses of these problems using duality theory. Our duality theory for the Domain-Driven form, which accepts both conic and non-conic constraints, lets us determine and certify statuses of a problem as rigorously as the best approaches for conic formulations (which have been demonstrably very efficient in this context). We analyze the performance of a class of infeasible-start primal-dual algorithms for the Domain-Driven form in returning the certificates for the defined statuses. Our iteration complexity bounds for this more practical Domain-Driven form match the best ones available for conic formulations. At the end, we propose some stopping criteria for practical algorithms based on insights gained from our analyses.

我们研究以领域驱动形式给出的凸优化问题的几何形状，并使用对偶理论对这些问题的可能状态进行分类。我们的领域驱动形式的对偶理论接受圆锥和非圆锥约束，让我们能够像圆锥公式的最佳方法一样严格地确定和证明问题的状态（在这种情况下，这已经证明非常有效）。我们分析了域驱动形式的一类不可行开始原始对偶算法在返回定义状态的证书时的性能。我们对这种更实用的领域驱动形式的迭代复杂度边界与可用于圆锥公式的最佳形式相匹配。最后，我们根据从我们的分析中获得的见解为实用算法提出了一些停止标准。