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On the Global Minimization of the Value-at-Risk

Jong-Shi Pang, Sven Leyffer

TL;DR

This paper tackles the nonconvex global minimization of VaR in a scenario-based setting by recasting VaR minimization as a linear program with equilibrium constraints (LPEC). It develops a suite of tight upper and lower bounds—via LP/NLP reformulations, convex-hull relaxations, and branching strategies—and embeds them in a branch-and-cut framework to certify global optimality of VaR solutions. A numerical example demonstrates how an initial upper bound from CVaR, refined bounds, and a compact branch-and-cut tree achieve proven global optimality, while a smoothing approach provides a convergent approximation pathway for VaR minimization. The work suggests that such tailored global optimization techniques can extend to broader classes of MPEC/LPEC problems, offering practical guarantees beyond local solvers.

Abstract

In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined bounding procedures can be used to compute the latter value to global optimality. A numerical example is provided to illustrate the methodology.

On the Global Minimization of the Value-at-Risk

TL;DR

This paper tackles the nonconvex global minimization of VaR in a scenario-based setting by recasting VaR minimization as a linear program with equilibrium constraints (LPEC). It develops a suite of tight upper and lower bounds—via LP/NLP reformulations, convex-hull relaxations, and branching strategies—and embeds them in a branch-and-cut framework to certify global optimality of VaR solutions. A numerical example demonstrates how an initial upper bound from CVaR, refined bounds, and a compact branch-and-cut tree achieve proven global optimality, while a smoothing approach provides a convergent approximation pathway for VaR minimization. The work suggests that such tailored global optimization techniques can extend to broader classes of MPEC/LPEC problems, offering practical guarantees beyond local solvers.

Abstract

In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined bounding procedures can be used to compute the latter value to global optimality. A numerical example is provided to illustrate the methodology.

Paper Structure

This paper contains 14 sections, 7 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The VaR Minimization Problem
  3. An LPEC Formulation
  4. Upper and Lower Bounds
  5. LP Upper Bounds
  6. NLP Upper Bounds
  7. LP Lower Bounds
  8. Convex Hull Relaxations
  9. Verifying Optimality by Branch-and-Cut
  10. Numerical Example
  11. Finding an Upper Bound
  12. Branching to Verify Global Optimality
  13. Approximation by Smoothing
  14. Conclusion

Key Result

Proposition 2.1

Let $X$ be a compact polyhedron in $\Re^n$. The LPEC (eq:VaR MPEC) attains a finite minimum objective value.

Figures (1)

  • Figure 1: Branch-and-bound tree for the numerical example

Theorems & Definitions (7)

  • Proposition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Lemma 6.1
  • Proposition 6.2