Operator Theory Arising from Systems Engineering

优化是现代技术最关键的领域之一，因为设计者总是试图最小化成本或最大化性能、安全性、产量等。它可以被认为是两个部分：凸优化和非凸优化。凸函数的概念是用一个杯子来说明的；它有唯一的最低点(最小值)，而对于非凸函数，人们会认为有许多山谷的山脉，因此有许多最低点(局部极小值)。计算机算法擅长找到一个(甚至几个)局部极小值，但一个主要的悬而未决的问题是：在所有的局部极小值中，找到最低的(全局)。对于凸问题，所有局部最小值都是全局的，这意味着计算机运行不会报告错误的最小值。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。在计算机上可解的凸优化问题主要有两类：经典线性规划问题和(近二十年来)更广泛应用的线性矩阵不等式(LMI)问题。这个项目涉及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？与合作者一起，国际和平组织为这个问题勾勒出了路线图，并进行了确认。人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开发一种有助于系统化这一点的理论。一个特别值得关注的问题是变量的变化，以将非凸问题转化为LMI。另一种是用凸集逼近集合。此外，PI的小组是公共软件(称为NCAlgebra)的主要提供者，用于在数学中执行常规的非交换代数计算。NCAlgebra是在为所提出的研究做实验的过程中发展起来的。经典实代数几何发展了(交换)多项式的理论，其中很大一部分涉及基于实数元组的计算的不等式。这个项目的很大一部分涉及非对易多项式及其在矩阵元组(所有大小)上求值时的性质。这种新的(自由的)非对易实代数几何通常比经典的实代数几何严格得多。虽然看到经典结构如何转移到自由的实代数几何是追求的一部分，但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、实现自由凸性的变量变换、自由凸包和自由膨胀理论都是数学丰富的领域，涉及泛函分析、最优化理论、代数和几个复变量。这个项目还研究了与其他学科的相互作用，如自由概率，以及主要与所谓的线性矩阵不等式有关的交换主题。

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.