[Paper Review] Relative Robust Portfolio Optimization

Raphael Hauser, V. Krishnamurthy+1 more|arXiv (Cornell University)|May 1, 2013

Risk and Portfolio Optimization23 references3 citations

TL;DR

This paper introduces a relative robust portfolio optimization framework that addresses uncertainty in mean-variance portfolio selection by minimizing maximum regret relative to scenario-specific optimal portfolios, rather than worst-case absolute performance. It provides tractable conic optimization-based inner and outer approximations for relative robust problems under polytopic and ellipsoidal uncertainty, enabling practical implementation while preserving theoretical guarantees for key asset management scenarios.

ABSTRACT

Considering mean-variance portfolio problems with uncertain model parameters, we contrast the classical absolute robust optimization approach with the relative robust approach based on a maximum regret function. Although the latter problems are NP-hard in general, we show that tractable inner and outer approximations exist in several cases that are of central interest in asset management.

Motivation & Objective

• To address the limitations of classical absolute robust optimization in portfolio management, which focuses on worst-case objective values and may unduly penalize decisions due to extreme scenarios.
• To develop a relative robust optimization framework based on regret minimization, where performance is evaluated relative to the optimal solution under each scenario, better aligning with real-world investment performance evaluation.
• To provide computationally tractable approximations for relative robust portfolio optimization problems, which are generally NP-hard, especially in continuous optimization settings.
• To extend the applicability of relative robust optimization beyond discrete problems to continuous quadratic and conic optimization problems relevant to asset management.
• To establish sufficient and necessary conditions for the tractability of relative robust formulations using conic programming and convex analysis, particularly under polytopic and ellipsoidal uncertainty sets.

Proposed method

• Formulates the relative robust portfolio optimization problem as a min-max regret problem, where regret is defined as the difference between the portfolio's objective value and the optimal value under each scenario.
• Uses conic optimization techniques to transform the three-level relative robust problem into a single-level deterministic problem by leveraging duality and convex analysis.
• Applies the S-lemma and generalized S-lemma to derive sufficient conditions for the feasibility of the relative robust problem, particularly for quadratic and second-order cone constraints.
• Derives inner and outer approximations of the relative robust solution set using convex hulls of uncertainty sets (polytopic) and ellipsoidal uncertainty sets, enabling efficient computation.
• Employs the Schur complement and matrix decomposition techniques to reformulate the problem into a form suitable for second-order cone programming (SOCP) or semidefinite programming (SDP).
• Introduces a lifting technique to embed the original problem into a higher-dimensional space where the relative robustness condition becomes a membership in a convex cone, enabling tractable solution via conic optimization.

Experimental results

Research questions

• RQ1 How can relative robust optimization be adapted to continuous portfolio optimization problems, particularly mean-variance models with uncertain parameters?
• RQ2 What are the computational challenges of relative robust portfolio optimization, and can they be mitigated through tractable approximations?
• RQ3 Under what uncertainty structures (e.g., polytopic, ellipsoidal) does the relative robust portfolio problem admit efficient conic programming reformulations?
• RQ4 Can sufficient and necessary conditions be derived for the tractability of relative robust portfolio problems using convex analysis and conic programming?
• RQ5 How does the relative robust approach compare to classical absolute robust and standard mean-variance optimization in terms of performance and robustness?

Key findings

• The relative robust portfolio optimization problem, though NP-hard in general, admits tractable inner and outer approximations under polytopic and ellipsoidal uncertainty sets.
• For polytopic uncertainty with up to one or two vertices (m_g ∈ {0,1}), the relative robust problem reduces to a set of second-order cone programs (SOCPs), enabling efficient solution via interior-point methods.
• The paper establishes that the relative robust formulation is equivalent to a conic feasibility problem involving the sum of positive semidefinite matrices and specific cone-generated matrices, enabling exact characterization under mild conditions.
• The use of the S-lemma and generalized S-lemma allows the derivation of sufficient conditions for the feasibility of the relative robust problem, which are also necessary when the number of uncertainty vertices is small.
• The proposed method provides a practical framework for asset managers to construct portfolios that are robust not against worst-case absolute performance, but against relative underperformance compared to scenario-specific optimal portfolios.
• Theoretical results show that the relative robust approach leads to smoother solution maps and better out-of-sample performance than standard mean-variance optimization, especially under parameter uncertainty.