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An Alternating Direction Method of Multipliers for Utility-based Shortfall Risk Portfolio Optimization

Rufeng Xiao, Zhiping Li, Rujun Jiang

TL;DR

A fast and scalable algorithm for the UBSR-based portfolio optimization problem that achieves global superlinear convergence with enhanced numerical stability under mild regularity conditions and a substantial speedup over state-of-the-art solvers.

Abstract

Utility-based shortfall risk (UBSR), a convex risk measure sensitive to tail losses, has gained popularity in recent years. However, research on computational methods for UBSR optimization remains relatively scarce. In this paper, we propose a fast and scalable algorithm for the UBSR-based portfolio optimization problem. Leveraging the Sample Average Approximation (SAA) framework, we reformulate the problem as a block-separable convex program and solve it efficiently via the alternating direction method of multipliers (ADMM). In the high-dimensional setting, a key challenge arises in one of the subproblems -- a projection onto a nonlinear feasibility set defined by the shortfall-risk constraint. We propose two semismooth Newton algorithms to solve this projection subproblem. The first algorithm directly applies a semismooth Newton iteration to the Karush-Kuhn-Tucker (KKT) system of the projection problem. The second algorithm employs an implicit function transformation of semismooth functions to reduce the problem to a univariate equation involving the Lagrange multiplier and achieves global superlinear convergence with enhanced numerical stability under mild regularity conditions. Theoretical convergence guarantees of the proposed algorithms are established, and numerical experiments demonstrate a substantial speedup over state-of-the-art solvers, particularly in high-dimensional regimes.

An Alternating Direction Method of Multipliers for Utility-based Shortfall Risk Portfolio Optimization

TL;DR

A fast and scalable algorithm for the UBSR-based portfolio optimization problem that achieves global superlinear convergence with enhanced numerical stability under mild regularity conditions and a substantial speedup over state-of-the-art solvers.

Abstract

Utility-based shortfall risk (UBSR), a convex risk measure sensitive to tail losses, has gained popularity in recent years. However, research on computational methods for UBSR optimization remains relatively scarce. In this paper, we propose a fast and scalable algorithm for the UBSR-based portfolio optimization problem. Leveraging the Sample Average Approximation (SAA) framework, we reformulate the problem as a block-separable convex program and solve it efficiently via the alternating direction method of multipliers (ADMM). In the high-dimensional setting, a key challenge arises in one of the subproblems -- a projection onto a nonlinear feasibility set defined by the shortfall-risk constraint. We propose two semismooth Newton algorithms to solve this projection subproblem. The first algorithm directly applies a semismooth Newton iteration to the Karush-Kuhn-Tucker (KKT) system of the projection problem. The second algorithm employs an implicit function transformation of semismooth functions to reduce the problem to a univariate equation involving the Lagrange multiplier and achieves global superlinear convergence with enhanced numerical stability under mild regularity conditions. Theoretical convergence guarantees of the proposed algorithms are established, and numerical experiments demonstrate a substantial speedup over state-of-the-art solvers, particularly in high-dimensional regimes.
Paper Structure (25 sections, 16 theorems, 63 equations, 2 figures, 5 tables, 6 algorithms)

This paper contains 25 sections, 16 theorems, 63 equations, 2 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Utility-Based Shortfall Risk
  4. Preliminaries on G-Semismooth Implicit Function Theorems
  5. ADMM for UBSR
  6. Problem Formulation
  7. ADMM for UBSR Optimization
  8. Projection onto the set Z
  9. Projection Problem Formulation
  10. A Semismooth Newton Method for Solving KKT System
  11. A G-semismooth Method for Solving a Univariate Equation
  12. Semismooth Newton Method for G-subproblem
  13. Properties of H(rho) and Convergence Results
  14. Global Convergence for Two Common Loss Functions
  15. Numerical Experiments
  16. ...and 10 more sections

Key Result

Lemma 2.4

Let $\boldsymbol{G}: \mathbb{R}^n \to \mathbb{R}^m$ and $\boldsymbol{F}: \mathbb{R}^m \to \mathbb{R}^k$ be G-semismooth functions with respect to $\mathcal{K}_{\boldsymbol{G}}$ and $\mathcal{K}_{\boldsymbol{F}}$, respectively. Assume that both $\mathcal{K}_{\boldsymbol{G}}$ and $\mathcal{K}_{\boldsy is also compact-valued and upper semicontinuous. Let $\boldsymbol{H} = \boldsymbol{F} \circ \boldsy

Figures (2)

  • Figure 1: Comparison of the problem dimensions and computational time for solving projection problem \ref{['p:projection on Z']} using different methods under different loss functions and parameter settings.
  • Figure 2: Cumulative returns comparison (out-of-sample).

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 23 more