Optimization over Positive Polynomials and Moment Cones: an Algorithmic Study with Applications in Approximation Theory, Regression and Data Visualization

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In this project the problem of approximating unknown functions with some shape constraints is being investigated. As a central tool properties of a class of functions known as positive polynomials are being investigated and utilized. The problem of approximation of unknown functions with shape constraints has many potential applications some of which is being developed and formulated in the context of this project. A sample of problems being investigated includes: shape constrained regression in statistics with applications in economics and finance (for instance estimating demand curve which is necessarily a decreasing function, or utility curves which are convex and decreasing); estimation of probabilities or odds of events based on some knowledge of their moments or frequency samples; visualization of data in the form of space curves and surfaces with shape constraints such as direction of curvatures or convexity. The study is related to an important and relatively new area called semidefinite programming, a field of mathematical optimization theory. While the problem of finding best positive polynomials can be cast as a semidefinite programming problem, the process may not be the most efficient way. In this project tailor made techniques are being developed to deal directly with positive polynomials leading to more efficient methods than using semidefinite programming methods directly.In this project two mathematical objects are being investigated. One is called "positive polynomials" and the other "moment cones". These objects have applications in several areas including statistics, economics, finance, various areas of engineering and two and three dimensional computer visualization of data. The main application of the objects of this study is in approximation and computer representation of behavior of unknown objects. The new problem that is being investigated is the imposition of "shape constraints" on the unknown phenomena and requiring that the computed approximation conforms to these constraints. For instance, in econometrics estimating the demand curve in some market from a scatter of sample data may yield an approximate demand curve that does not uniformly move down as price goes up. The properties of "positive polynomials" and "moment cones" are investigated to assist in computer shape constrained approximation of unknown objects that arise in fields such as economics, engineering and data visualization.