Theory and Efficient Algorithms for Hard, Large Scale, Numerical Optimization

我的研究重点将是对困难问题进行适当的建模，以及为大规模、困难的优化问题设计和实现高效和健壮的数值算法。我将要处理的问题出现在许多重要的应用中，例如分子构象(MC)、传感器网络定位(SNL)、逆成像和机器学习。特别是，这些问题中的许多都是在困难的组合优化问题的松弛下出现的。在许多情况下，通常的建模方法会导致大规模和不适定的问题。因此，它们很难在数值上求解。与其成为劣势，人们往往可以利用这种不适定性，既得到一个稳定的问题，又得到一个规模较小的问题。特别是，对于像SNL这样的问题，人们可以利用隐藏的简并性来高精度地解决巨大的问题。我计划将这一技术应用于带有噪声数据的MC问题以及蛋白质设计问题。 我使用的技术涉及连续优化、非线性规划(NLP)，特别是半定规划(SDP)。对于SDP，我的研究贡献涉及理论、算法和应用，即它们包括强对偶结果、稳定算法和各种应用的松弛研究。我为标准形式线性规划(LP)模型设计的一些代码已经在Maple中实现。我计划为更一般的LP模型实现算法，包括变量的上下限。此外，我还计划实现解决更一般NLP问题的代码。对于NLP问题，我使用广义信赖域问题的算法来求解大规模无约束极小化问题，并使用稳定的序列二次规划方法来求解一般的NLP问题。这些实施的成功意味着我这个领域的研究人员将有机会获得稳定、高精度的算法。

The focus of my research will be the proper modelling of hard problems, and the design and implementation of efficient and robust numerical algorithms for large scale, hard, optimization problems. The problems I will deal with arise in many important applications, e.g. molecular conformation (MC), sensor network localization (SNL), inverse imaging and machine learning. In particular, many of these problems arise in the relaxations of hard combinatorial optimization problems. In many instances, the usual modelling approaches result in problems that are both large scale and ill-posed. Therefore, they are hard to solve numerically. Rather than being a disadvantage, one can often take advantage of the ill-posedness to get both a stable problem and one that is smaller in size. In particular, for problems such as SNL one can solve huge problems to high accuracy by exploiting the hidden degeneracy. I plan on applying this technique to MC problems with noisy data as well as to protein design problems. The techniques that I use involve continuous optimization, nonlinear programming (NLP) and in particular, semidefinite programming (SDP). For SDP, my research contributions involve theory, algorithms, and applications, i.e., they include strong duality results, stable algorithms, and the study of relaxations for various applications. Some of the codes that I have designed for the standard form Linear Programming (LP) model have already been implemented in MAPLE. I plan to implement algorithms for more general LP models that include both upper and lower bounds on the variables. As well I plan on implementing codes that solve more general NLP problems. For the NLP problems, I use algorithms for generalized trust region problems to solve large scale unconstrained minimization, as well as solve general NLP using stable sequential quadratic programming methods. The success of these implementations means that researchers in my field will have access to stable, high accuracy, algorithms.