Operator Theory Arising from Systems Engineering

优化是现代技术最关键的领域之一，因为设计师总是试图最小化成本或最大化性能，安全性，产量等。它可以分为两部分：凸优化和非凸优化。凸函数的概念用一个杯子来说明；它有唯一的最低点（最小值），而对于非凸问题，人们会想到一个有许多山谷的山脉，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）局部最小值，但一个主要的开放问题是：在所有的局部最小值中，找到最小的（全局）最小值。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假最小值。当然，在技术领域，变量的数量是巨大的，所以设计师所能获得的只是代数公式（而不是图片）；因此，杯和山的比喻简单得令人误解。在计算机上可解决的凸优化问题主要有两类：经典线性规划和（近二十年来）更广泛应用的线性矩阵不等式（lmi）。该项目涉及LMI的许多方面，包括LMI技术的范围：用LMI可以处理的问题是凸的，但反过来，哪些凸问题可以用LMI处理？PI与合作者一起勾画了这个问题的路线图，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换为凸问题目前是通过特殊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。一个特别关注的问题是改变变量将非凸问题转换为lmi。另一种是用凸集逼近一个集合。此外，PI的小组是向公众提供在Mathematica中执行一般非交换代数计算的软件（称为NCAlgebra）的主要提供者。NCAlgebra是在为拟议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一种（交换）多项式理论，其中大部分涉及基于实数元组上对多项式求值的不等式。这个项目的很大一部分涉及非交换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）非交换实代数几何通常比经典实代数几何表现得更加严格。当看到经典结构如何转化为自由的真实代数几何是追求的一部分时，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、改变变量以实现自由凸性、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、代数和几个复杂变量的混合。该项目还研究了与其他学科的相互作用，如自由概率以及主要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.