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A generalized moment approach to sharp bounds for conditional expectations

Wouter J. E. C. van Eekelen

TL;DR

This paper proposes an adapted version of the generalized moment problem which deals with conditional expectations when moment information of the underlying distribution and the random event conditioned upon are given, and derives computationally tractable mathematical programs for distributionally robust optimization with side information.

Abstract

In this paper, we address the problem of bounding conditional expectations when moment information of the underlying distribution and the random event conditioned upon are given. To this end, we propose an adapted version of the generalized moment problem which deals with this conditional information through a simple transformation. By exploiting conic duality, we obtain sharp bounds that can be used for distribution-free decision-making under uncertainty. Additionally, we derive computationally tractable mathematical programs for distributionally robust optimization (DRO) with side information by leveraging core ideas from ambiguity-averse uncertainty quantification and robust optimization, establishing a moment-based DRO framework for prescriptive stochastic programming.

A generalized moment approach to sharp bounds for conditional expectations

TL;DR

This paper proposes an adapted version of the generalized moment problem which deals with conditional expectations when moment information of the underlying distribution and the random event conditioned upon are given, and derives computationally tractable mathematical programs for distributionally robust optimization with side information.

Abstract

In this paper, we address the problem of bounding conditional expectations when moment information of the underlying distribution and the random event conditioned upon are given. To this end, we propose an adapted version of the generalized moment problem which deals with this conditional information through a simple transformation. By exploiting conic duality, we obtain sharp bounds that can be used for distribution-free decision-making under uncertainty. Additionally, we derive computationally tractable mathematical programs for distributionally robust optimization (DRO) with side information by leveraging core ideas from ambiguity-averse uncertainty quantification and robust optimization, establishing a moment-based DRO framework for prescriptive stochastic programming.
Paper Structure (16 sections, 13 theorems, 120 equations, 5 figures)

This paper contains 16 sections, 13 theorems, 120 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A duality framework for generalized conditional-bound problems
  3. Problem statement
  4. An equivalent problem and its dual
  5. Strong duality of the generalized conditional-bound problem
  6. Tight bounds for conditional expectations
  7. Simple examples for mean-dispersion information
  8. A robust pricing example
  9. Numerical bounds
  10. Distributionally robust optimization with side information
  11. Contextual DRO with mean-dispersion information
  12. Some examples for mean-dispersion information
  13. Newsvendor example
  14. Conclusions and outlook
  15. Proofs
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose that problem eq:condmomentprob has a finite optimal value. Then problem eq:condmomentprob is equivalent to That is, the optimal value $\tau^*$ agrees with that of eq:generalmomentproblem, and the suprema in both eq:generalmomentproblem and eq:altern are achieved exactly by the same extremal distribution $\mathbb{P}^*$, or achieved asymptotically by an identical sequence of distributions $

Figures (5)

  • Figure 1: $M_1(x)$ and $M_2(x)$
  • Figure 2: Tight bounds for conditional tail probability for ambiguity sets matching the uniform distribution on $[0,5]$
  • Figure 3: Tight bounds for conditional expectation for ambiguity sets matching the moments and properties of the standard normal distribution
  • Figure 4: Newsvendor cost bounds for different order quantities with dependent and independent demand
  • Figure 5: $D_1(x)$, $D_2(x)$, and $D_3(x)$

Theorems & Definitions (25)

  • Lemma 1
  • Theorem 1: Strong conic duality
  • proof
  • Theorem 2: Fundamental theorem for conditional expectations
  • proof
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 15 more