On Accelerating Large-Scale Robust Portfolio Optimization

TL;DR

This work tackles the computational burden of large-scale robust portfolio optimization under distributional uncertainty. It extends the supporting hyperplane approximation to accommodate a broad class of additively separable utilities and turnover costs, formulating a robust linear program that remains scalable for hundreds of assets. A convex polyhedral ambiguity set is paired with a dual reformulation, enabling efficient solution and rigorous approximation error analysis that partitions into $x$- and $c$-directions. Empirical validation on a S&P 500 cross-section demonstrates drastic reductions in runtime (from thousands to seconds) with robust out-of-sample performance, validating the approach's practicality for real-world, large-scale portfolio management under uncertainty.

Abstract

Solving large-scale robust portfolio optimization problems is challenging due to the high computational demands associated with an increasing number of assets, the amount of data considered, and market uncertainty. To address this issue, we propose an extended supporting hyperplane approximation approach for efficiently solving a class of distributionally robust portfolio problems for a general class of additively separable utility functions and polyhedral ambiguity distribution set, applied to a large-scale set of assets. Our technique is validated using a large-scale portfolio of the S&P 500 index constituents, demonstrating robust out-of-sample trading performance. More importantly, our empirical studies show that this approach significantly reduces computational time compared to traditional concave Expected Log-Growth (ELG) optimization, with running times decreasing from several thousand seconds to just a few. This method provides a scalable and practical solution to large-scale robust portfolio optimization, addressing both theoretical and practical challenges.

Figures (4)

• Figure 1: Trading Performance of Four Portfolios: SPY, EW, ELG and HYP with Ambiguity Constant $\gamma = 0$, Turnover Cost Limit $c_{\max} = 1.5$, and Various Cost Rates $c \in \{0, 0.001, 0.005\}$.
• Figure 2: Account Values of HYP under Different $\gamma$, without Diversified Holding Constraint.
• Figure 3: Expected Return of HYP Portfolio with Various $\gamma$.
• Figure 4: Diversification Effects via Ambiguity Constant in HYP: The Maximum Portfolio Weight $\max_i K_i^{Y}(t)$ Versus $\gamma$ where $\max_i K_i^{Y}(t)$ is the Maximum Portfolio Weight of the $Y \in \{2021, 2022, 2023\}$ Year.

Theorems & Definitions (44)

• Remark 2.1: Assumption of Return Model
• Remark 2.2: Zero Cost Case
• Lemma 2.1: Survivability Condition
• proof
• Definition 2.1: Turnover Rate
• Remark 2.3: Turnover Rate Constraint
• Definition 2.2: Totality of the Trading Constraints
• Remark 2.4: Convexity and Compactness of $\mathcal{K}$
• Theorem 2.1: An Equivalent Distributional Robust Optimization Problem
• proof