Mathematical Optimization: Theory and Algorithms

我的研究项目旨在为计算机科学中处理的问题找到可处理的数学模型，然后专注于这些问题的高效和鲁棒算法（解决方法）的设计，数学分析和实现。这些问题出现在信息技术（包括量子信息和计算）、制造业、交通运输、规划、经济、金融以及服务业等许多应用领域。研究计划的重点是通过数学优化问题尽可能准确地（和合理地）建模这些问题及其数学概括。这些数学优化问题通常属于一个特殊的子类（即具有额外的特殊结构）：(a) 0,1混合整数规划，或(b)寻求低秩解的半定优化问题，或(c)存在离散变量的半定优化问题，或(d)寻求最高秩解的半定优化问题，或(e)由多项式方程和不等式定义的优化问题（可能是非凸的）。由于凸优化问题形成了一个非常广泛的可处理的数学优化问题类别（在合理的假设下，这样的问题，当适定时，可以在多项式时间内以任意精度求解），下一步是构造一个可处理的凸近似来处理原始的，困难的数学优化问题。从理论的角度来看，这种方法为设计原始对偶算法提供了一个框架。然后，这个框架与近似最优解一起导致最优性证明。对于困难的问题，我们只能希望得到近似最优性的证明，因此，对于这些问题，我们关注的是有效的近似算法。本研究项目将推进一阶和高阶复合算法的设计、研究和实现，这些算法关注大数据机制并适应给定的数据实例。这些算法将结合一阶算法的所需特征：低内存要求，2。每次迭代的低复杂度，2。可分布性、4。可平行性;二阶和高阶算法：更快的全球收敛，6。更快的局部收敛，7。8.高精度解决方案；鲁棒性。只要可行，最终实现的源代码以及用于计算测试和基准测试的数据将在网络上提供。

My research program aims to find tractable mathematical models for problems treated within computer science, and then focuses on the design, mathematical analysis and implementation of efficient and robust algorithms (solution methods) for such problems. These problems arise in many application areas such as information technology (including quantum information and computing), manufacturing, transportation, planning, economics, finance, as well as service sectors. The research program focuses on modelling these problems and their mathematical generalizations as accurately as possible (and reasonable) by mathematical optimization problems. These mathematical optimization problems will typically lie in a special subclass (i.e., with additional special structures) of: (a) 0,1 mixed integer programming, or (b) semidefinite optimization problems where we seek low-rank solutions, or (c) semidefinite optimization problems where there are discrete variables, or (d) semidefinite optimization problems where we seek highest rank solutions,or (e) optimization problems defined by (possibly nonconvex) polynomial equations and inequalities. Since convex optimization problems form a very wide class of tractable mathematical optimization problems (under reasonable assumptions, such problems, when well-posed, can be solved to arbitrary accuracy in polynomial-time), the next step is the construction of a tractable convex approximation to the original, hard mathematical optimization problem. From a theoretical viewpoint, this approach provides a framework to design primal-dual algorithms. This framework then leads to, together with approximately optimal solutions, certificates of optimality. For hard problems, we can only hope for certificates of approximate optimality, hence for those, we focus on efficient approximation algorithms. This research program will advance the design, study and implementation of composite first--order and higher--order algorithms which focus on the big-data regime and adapt to the given data instance. These algorithms will combine the desired features of first--order algorithms: 1. low memory requirements, 2. low complexity per iteration, 3. distributability, 4. parallelizability; with those of second--order and higher--order algorithms: 5. much faster global convergence, 6. much, much faster local convergence, 7. high accuracy solutions, 8. robustness. Whenever feasible, the source codes of the resulting implementations as well as the data used for computational test and benchmarking will be made available on the web.