Operator Theory Arising from Systems Engineering

优化是现代技术中最关键的领域之一，因为设计师总是试图最大限度地降低成本或最大限度地提高性能，安全性，产量等。凸函数的概念可以用一个杯子来说明;它有唯一的最低点（最小值），而对于非凸问题，人们会认为山脉有许多山谷，因此有许多最低点（局部最小值）。计算机算法擅长于找到一个（甚至几个）局部极小值，但一个主要的开放问题是：在所有的局部极小值中，找到最低的（全局）极小值。对于凸问题，所有的局部极小值都是全局的，这意味着计算机运行不会报告错误的极小值。当然，在技术中，变量的数量是巨大的，所以设计者所能得到的只是代数公式（而不是图片）;因此，杯子和山的比喻简单得令人误解。有两大类凸优化问题可以在计算机上解决：经典线性规划和（在过去的二十年中）更广泛应用的线性矩阵不等式（LMI）。这个项目涉及到LMI的许多方面，包括LMI技术的范围：可以用LMI处理的问题是凸的，但反过来，哪些凸问题可以用LMI处理？PI与合作者一起为这个问题勾画了一个路线图，并寻求确认。我们在线性系统工程和控制中看到的是矩阵未知数的问题。简化物理问题并将其转化为凸问题目前是通过特别的代数技巧来完成的。这个项目的一个主要目标是发展一个理论，将有助于系统化。一个特别关注的是变量的变化，将非凸问题转化为LMI。另一个是用凸集来近似一个集合。此外，PI的小组是向公众提供软件（称为NCAlgebra）的主要供应商，用于在Mathematica中执行一般非交换代数计算。经典的真实的代数几何发展了一种（可交换的）多项式的理论，其中大部分涉及基于对真实的数的元组求值的不等式。这个项目的一个很好的部分涉及非交换多项式和它们的性质时，评估元组的矩阵（所有大小）。这种新的（自由的）非交换的真实的代数几何往往比经典的真实的代数几何表现得更加严格。虽然看到经典结构如何运输到自由的真实的代数几何是追求的一部分，工程动机和高度刚性的结构开辟了新的问题类别。例如，自由凸性，变量的变化以实现自由凸性，自由凸包和自由膨胀理论是数学上丰富的领域，涉及泛函分析，优化理论，代数和几个复变量的混合。该项目还研究了与其他学科的相互作用，如自由概率以及与所谓的线性矩阵不等式相关的交换主题。

Optimization is one of the areas most critical to modern technology, since designers always try to minimize cost or maximize performance, safety, output, etc. It can be thought of in two parts: convex optimization and nonconvex optimization. The concept of a convex function is illustrated by a cup; it has unique lowest point (minimum), while for nonconvex problems one would think of a mountain range with many valleys, hence many lowest points (local minima). Computer algorithms are good at finding one (or even a few) of local minima, but a major open problem is this: out of all the local minima, find the lowest (global) one. For convex problems all local minima are global, which means that computer runs do not report a false minimum. Of course, in technology the number of variables is huge, so all that is available to the designer are algebraic formulas (not pictures); thus the cup and mountain metaphors are misleadingly simple. There are two major classes of convex optimization problems solvable on a computer: classical linear programing and (within the last twenty years) the more widely applicable linear matrix inequalities (LMIs). This project concerns many aspects of LMIs, including the scope of LMI techniques: problems treatable with LMIs are convex, but conversely, which convex problems are treatable with LMIs? With collaborators the PI has sketched out a roadmap for this problem and pursues its confirmation. What one sees in linear systems engineering and control are problems with matrix unknowns. Simplifying physical problems and converting them to convex ones is currently done by ad hoc algebraic tricks. A major goal in this project is to develop a theory that will help systematize this. A particular concern is changes of variables to convert nonconvex problems to LMIs. Another is approximating a set with a convex set. In addition, the PI's group is the main provider to the public of software (called NCAlgebra) for performing general noncommuting algebra calculations in Mathematica. NCAlgebra is developed in the course of doing experiments for the proposed research.Classical real algebraic geometry develops a theory of (commutative) polynomials and much of it concerns inequalities based on evaluating them on tuples of real numbers. A good part of this project concerns noncommutative polynomials and their properties when evaluated on tuples of matrices (of all sizes). This new (freely) noncommutative real algebraic geometry often behaves much more rigidly than classical real algebraic geometry. While seeing how classical structure transports to free real algebraic geometry is part of the pursuit, engineering motivation and the highly rigid structure opens up new classes of problems. For example, free convexity, change of variables to achieve free convexity, free convex hulls, and free dilation theory are mathematically rich areas involving mixtures of functional analysis, optimization theory, algebra, and several complex variables. Also studied in this project are interactions with other subjects such as free probability as well as commutative topics related mostly to so-called linear matrix inequalities.