Mathematical Optimization: Theory and Algorithms

我的研究计划旨在为计算机科学中处理的问题找到易于处理的数学模型，然后专注于此类问题的高效和健壮算法(求解方法)的设计、数学分析和实现。这些问题出现在许多应用领域，如信息技术(包括量子信息和计算)、制造、交通、规划、经济、金融以及服务部门。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)对这些问题及其数学概括进行建模。这些数学优化问题通常属于一个特殊的子类(即，具有附加的特殊结构)：(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)有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(可能非凸)多项式方程和不等式定义的优化问题。由于凸优化问题形成了一类非常容易处理的数学优化问题(在合理的假设下，当这类问题适定时，可以在多项式时间内求解到任意精度)，因此下一步是构造对原始的、困难的数学优化问题的易处理的凸近似。从理论上讲，该方法为原始-对偶算法的设计提供了一个框架。然后，该框架与近似最优解决方案一起产生最优证书。对于困难问题，我们只能寄希望于近似最优性证书，因此，对于这些问题，我们专注于高效的近似算法。该研究项目将推进一阶和高阶混合算法的设计、研究和实现，这些算法着眼于大数据机制，并适应于给定的数据实例。这些算法将结合一阶算法的期望特征：1.低存储需求，2.低每次迭代的复杂度，3.可分布性，4.并行化；与二阶和高阶算法的那些：5.更快的全局收敛，6.更快的局部收敛，7.高精度解，8.稳健性。只要可行，最终实现的源代码以及用于计算测试和基准的数据都将在网上提供。

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.