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Optimization with Multi-sourced Reference Information and Unknown Trust: A Distributionally Robust Approach

Yanru Guo, Ruiwei Jiang, Siqian Shen

TL;DR

This paper tackles optimization under parameter uncertainty when information comes from multiple data sources with varying reliability. It introduces a multi-reference distributionally robust optimization (MR-DRO) framework that builds a Wasserstein ambiguity set $B_{epsilon}(P_{HI}^{emp})$ via trust-weighted nonparametric data fusion and yields tractable LP reformulations. A dynamic trust update mechanism and the notion of probability dominance are developed to adapt source weights over time. Through resource allocation and portfolio optimization experiments, MR-DRO consistently outperforms single-source DRO, illustrating the practical value of incorporating dynamic trust in decision making under uncertainty.

Abstract

In problems that involve input parameter information gathered from multiple data sources with varying reliability, incorporating users' trust about different sources in decision-optimization models can potentially improve solution performance and reliability. In this work, we propose a novel multi-reference distributionally robust optimization (MR-DRO) framework, where the model inputs are uncertain and their probability distributions can be statistically inferred from multiple data sources. Via nonparametric data fusion, we construct a Wasserstein ambiguity set to minimize the worst-case expected value of a stochastic objective function, accounting for both uncertainty and unknown reliability of information sources. We reformulate the MR-DRO model as a linear program given linear objective and constraints in the original problem. We also incorporate a dynamic trust update mechanism that adjusts the trust for each source based on its performance over time. In addition, we introduce the concept of probability dominance to identify sources with dominant trust. Via solving instances of resource allocation and portfolio optimization, we demonstrate the effectiveness of the trust-informed MR-DRO approach compared to traditional optimization frameworks relying on a single data source. Our results highlight the significance of integrating (dynamic) user trust in decision making under uncertainty, particularly when given diverse and potentially conflicting input data.

Optimization with Multi-sourced Reference Information and Unknown Trust: A Distributionally Robust Approach

TL;DR

This paper tackles optimization under parameter uncertainty when information comes from multiple data sources with varying reliability. It introduces a multi-reference distributionally robust optimization (MR-DRO) framework that builds a Wasserstein ambiguity set via trust-weighted nonparametric data fusion and yields tractable LP reformulations. A dynamic trust update mechanism and the notion of probability dominance are developed to adapt source weights over time. Through resource allocation and portfolio optimization experiments, MR-DRO consistently outperforms single-source DRO, illustrating the practical value of incorporating dynamic trust in decision making under uncertainty.

Abstract

In problems that involve input parameter information gathered from multiple data sources with varying reliability, incorporating users' trust about different sources in decision-optimization models can potentially improve solution performance and reliability. In this work, we propose a novel multi-reference distributionally robust optimization (MR-DRO) framework, where the model inputs are uncertain and their probability distributions can be statistically inferred from multiple data sources. Via nonparametric data fusion, we construct a Wasserstein ambiguity set to minimize the worst-case expected value of a stochastic objective function, accounting for both uncertainty and unknown reliability of information sources. We reformulate the MR-DRO model as a linear program given linear objective and constraints in the original problem. We also incorporate a dynamic trust update mechanism that adjusts the trust for each source based on its performance over time. In addition, we introduce the concept of probability dominance to identify sources with dominant trust. Via solving instances of resource allocation and portfolio optimization, we demonstrate the effectiveness of the trust-informed MR-DRO approach compared to traditional optimization frameworks relying on a single data source. Our results highlight the significance of integrating (dynamic) user trust in decision making under uncertainty, particularly when given diverse and potentially conflicting input data.
Paper Structure (42 sections, 7 theorems, 61 equations, 9 figures, 7 tables, 3 algorithms)

This paper contains 42 sections, 7 theorems, 61 equations, 9 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Structure of the Paper
  4. Notation
  5. Models and Solution Approaches
  6. Nonparametric data fusion for multiple sources
  7. Multi-reference distributionally robust optimization (MR-DRO) model
  8. Tractable Reformulations of the MR-DRO Model
  9. Convex reduction
  10. Reformulations with regards to certain loss functions
  11. Dynamic Trust Update Process
  12. Min-max error trust update
  13. Exponential error trust update
  14. Variable-share error trust update
  15. Analyzing trust dynamics
  16. ...and 27 more sections

Key Result

Theorem 1

Given Assumption asp:convexity, then for any $\epsilon \geq 0$ the inner worst-case expectation eq:worst-case-expectation equals the optimal value of the finite convex program eq:DRO-finite-convex-program: Here, the conjugate of a function $f$ is defined as $f^{*}(\bm{v}):= \sup_{\bm{u} \in \text{dom }f}\langle \bm{v},\bm{u}\rangle - f(\bm{u})$. $\|\cdot\|_{*}$ is the dual norm of the norm used i

Figures (9)

  • Figure 1: Illustrating trust update process with method-dependent metrics
  • Figure 2: Prediction error distributions with the baseline setting for the resource allocation problem
  • Figure 3: Trust Update Process with (\ref{['fig:resource_trust_update_min_max']}) Min-max error trust update; (\ref{['fig:resource_trust_update_exponential']}) Exponential error trust update; (\ref{['fig:resource_trust_update_variable_share']}) Variable-share error trust update in the baseline setting for the resource allocation problem ($K_{\text{r}} = 4$, $H = 3$, $I = 200$).
  • Figure 4: Average loss of different methods as number of events increases for the resource allocation problem
  • Figure 5: Prediction error distributions for the resource allocation problem (varying distribution types)
  • ...and 4 more figures

Theorems & Definitions (12)

  • Example 1: Nonparametric data fusion
  • Theorem 1: Convex reduction
  • Theorem 2: Reformulation with piecewise affine loss functions
  • Theorem 3: Reformulation with separable affine objective functions
  • Example 2: Min-max error trust update
  • Example 3: Exponential error trust update
  • Example 4: Variable-share error trust update
  • Definition 1: Probability dominance, wrather1982probability
  • Theorem 4: Dominant trust under min-max error trust update for two sources
  • Theorem 5: Dominant trust under min-max error trust update for multiple sources
  • ...and 2 more