Theory of Mathematical Optimization for Delegated Portfolio Management

Author: Tao, Zijian

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Cvitanić, Jakša

Committee Members: Tamuz, Omer; Cvitanić, Jakša; Makarov, Nikolai G.; Sandomirskiy, Fedor

Option: Mathematics

DOI: 10.7907/km2b-er60

Abstract

We study the optimization problem of finding closed convex sets Γ ⊆ Rd containing the origin that minimize F(Γ) = ∑i=1k wi | θi/2 - pΓ(θi) | 2, where w1, ..., wk > 0, θ1, ..., θk in Rd are given, and pΓ(θi) are the closest points in Γ to θi, 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.

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