Theory of Mathematical Optimization for Delegated Portfolio Management

Citation

Tao, Zijian (2022) Theory of Mathematical Optimization for Delegated Portfolio Management. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/km2b-er60. https://resolver.caltech.edu/CaltechTHESIS:05272022-034901698

Abstract

We study the optimization problem of finding closed convex sets Γ ⊆ R^d containing the origin that minimize F(Γ) = ∑_i=1^k w_i | θ_i/2 - p_Γ(θ_i) | ², where w₁, ..., w_k > 0, θ₁, ..., θ_k in R^d are given, and p_Γ(θ_i) are the closest points in Γ to θ_i, i = 1, ..., k. This problem is motivated by the topic of delegated portfolio management in finance. In Chapter 2, we will explore this connection. To approach the problem, we first prove existence of a solution for the general problem. To further study properties of the solution, we next introduce the semidefinite programming relaxation, for which we have a first-order characterization of optimality. We then explore the question of exactness of this relaxation, which turns out to be equivalent to the notion of localizability: the shape optimization problem embedded in higher dimensions must have solutions in the original dimension. Finally, we present special cases for which localizability holds.

Thesis Files