Solving ill-posed conic optimization problems

(a)我们完成了以下主题的几个预印本：双曲锥的几何，幂锥的误差界和多面体锥的自对偶性。在我们的结果中，我们能够证明所有的双曲锥是顺从的，我们还研究了它们的自同构群在一定的条件下。对于幂锥，我们完全确定了它们的误差界和自同构。最后，我们发现，自对偶多面体锥可以完全检测到通过其松弛矩阵之一的正半定性，我们发现了一个令人惊讶的联系之间的松弛矩阵的不可约自对偶多面体锥和极端射线的双非负矩阵。(b)在对上一财政年度同行评审的论文进行修订后，其中几篇论文最终被优化和邻近领域的重要期刊接受。这包括论文在SIAM杂志上优化，SIAM杂志上应用代数和几何，SIAM杂志上矩阵分析，数学规划和计算数学的基础。(c)我们在网上和面对面的研讨会和会议上展示了我们的成果。还对澳大利亚和巴西的合作者进行了研究访问。

(a) We completed several preprints on the following topics: geometry of hyperbolicity cones, error bounds for power cones and self-duality of polyhedral cones. Among our results, we were able to show that all hyperbolicity cones are amenable and we also investigated their automorphism group under certain conditions. For power cones, we completely determined their error bounds and automorphisms. Finally, we showed that self-duality for a polyhedral cone can be completely detected through the positive semidefiniteness of one of its slack matrices and we showed a surprising connection between slack matrices of irreducible self-dual polyhedral cones and extreme rays of doubly nonnegative matrices.(b) Following the revision of papers that were in peer-review in the previous fiscal year, several of those papers were finally accepted at important journals in optimization and neighbouring areas. This includes papers in SIAM Journal on Optimization, SIAM Journal on Applied Algebra and Geometry, SIAM Journal on Matrix Analysis, Mathematical Programming and Foundations of Computational Mathematics.(c) We presented our results in workshops and conferences both online and in-person. There were also research visits to collaborators in Australia and Brazil.