Many convex optimization problems can be represented through conic extended formulations with auxiliary variables and constraints using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. Such extended formulations are often significantly larger and more complex than equivalent conic natural formulations, which can use a much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement tractable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. In this paper we introduce Hypatia, a highly-configurable open-source conic primal-dual interior point solver with a generic interface for exotic cones. Hypatia is written in Julia and accessible through JuMP, and currently implements around two dozen useful predefined cones (some with multiple variants). We define some of Hypatia's exotic cones, and for conic constraints over these cones, we analyze techniques for constructing equivalent representations using the standard cones. For optimization problems from a variety of applications, we introduce natural formulations using these exotic cones, and we show that the natural formulations are simpler and lower-dimensional than the equivalent extended formulations. Our computational experiments demonstrate the potential advantages, especially in terms of solve time and memory usage, of solving the natural formulations with Hypatia compared to solving the extended formulations with either Hypatia or MOSEK 9.
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
本文提出了MIQCR~（Billionnet，Elloumi and Lambert，2012）的改进算法MIQCR-CB。MIQCR是一种求解混合整数二次规划的方法，分为两个阶段：第一阶段通过求解半定问题$(SDP)$来确定具有凸目标函数的等效二次规划，并且在第二阶段中，通过标准求解器来求解等效规划。由于重新制定依赖于一个大规模的半定方案的解决方案，它是不听话的现有的半定求解器，已经为中等规模的问题。为了克服这一困难，我们提出了在MIQCR-CB次梯度法内的拉格朗日对偶框架求解$(SDP)$，大大加快了第一阶段。此外，该算法导致了一个更小的尺寸比原来的MIQCR方法，从而在更短的时间内解决第二阶段得到的重新制定的问题。我们提出了广泛的计算结果，以显示我们的算法的效率。
Solving Conic Programs over Sparse Graphs using a Variational Quantum Approach: The Case of the AC Optimal Power Flow
Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47906, USA（电子与计算机工程系，普渡大学）；School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA（电子与计算机工程系，康奈尔大学）
A Practical GPU-Enhanced Matrix-Free Primal-Dual Method for Large-Scale Conic Programs
Purdue University（普渡大学）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. 米尔顿·斯图尔特工业与系统工程学院）；Antai School of Economics and Management, Shanghai Jiao Tong University（上海交通大学安泰经济与管理学院）；Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程系）
