Algebraic Methods in Optimization
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Author(s)
Shu, Kevin
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Abstract
This thesis broadly concerns the usage of techniques from algebra, the study of higher order structures in mathematics, toward understanding difficult optimization problems. Of particular interest will be optimization problems related to systems of polynomial equations, algebraic invariants of topological spaces, and algebraic structures in convex optimization.
We will discuss various concrete examples of these kinds of problems. Firstly, we will describe new constructions for a class of polynomials known as hyperbolic polynomials which have connections to convex optimization. Secondly, we will describe how we can use ideas from algebraic geometry, notably the study of Stanley-Reisner varieties to study sparse structures in semidefinite programming. This will lead to quantitative bounds on some approximations for sparse problems and concrete connections to sparse linear regression and sparse PCA. Thirdly, we will use methods from algebraic topology to show that certain optimization problems on nonconvex topological spaces can be turned into convex problems due to a phenomenon known as `hidden convexity'. Specifically, we give a sufficient condition for the image of a topological space under a continuous map to be convex, and give a number of examples of this phenomena with practical importance. This unifies and generalizes a number of existing results. Finally, we will describe how to use techniques inspired by the sum of squares method to find new variants of gradient descent which converge faster than typical gradient descent on smooth convex problems.
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2024-04-26
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Dissertation