Operator Theory Arising from Systems Engineering

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.