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SOLID: a Framework of Synergizing Optimization and LLMs for Intelligent Decision-Making

Yinsheng Wang, Tario G You, Léonard Boussioux, Shan Liu

TL;DR

SOLID addresses the challenge of leveraging unstructured information in decision-making by fusing optimization with LLM reasoning. It uses an ADMM-inspired coordination with dual prices $\lambda_{opt}$ and $\lambda_{llm}$ and deviation penalties to drive consensus on a shared decision variable $x=z$, minimizing a sum of subproblem objectives via the augmented Lagrangian $\mathcal{L}_{\rho}$ to achieve convergence under convexity. The case study on portfolio optimization with historical prices and unstructured news demonstrates improved risk-adjusted returns and robust convergence across diverse LLMs, validating the synergy between precise optimization and contextual reasoning. The work offers a modular, privacy-preserving framework and prompts design guidance, with potential applicability to other decision-making domains requiring integration of structured and unstructured data.

Abstract

This paper introduces SOLID (Synergizing Optimization and Large Language Models for Intelligent Decision-Making), a novel framework that integrates mathematical optimization with the contextual capabilities of large language models (LLMs). SOLID facilitates iterative collaboration between optimization and LLMs agents through dual prices and deviation penalties. This interaction improves the quality of the decisions while maintaining modularity and data privacy. The framework retains theoretical convergence guarantees under convexity assumptions, providing insight into the design of LLMs prompt. To evaluate SOLID, we applied it to a stock portfolio investment case with historical prices and financial news as inputs. Empirical results demonstrate convergence under various scenarios and indicate improved annualized returns compared to a baseline optimizer-only method, validating the synergy of the two agents. SOLID offers a promising framework for advancing automated and intelligent decision-making across diverse domains.

SOLID: a Framework of Synergizing Optimization and LLMs for Intelligent Decision-Making

TL;DR

SOLID addresses the challenge of leveraging unstructured information in decision-making by fusing optimization with LLM reasoning. It uses an ADMM-inspired coordination with dual prices and and deviation penalties to drive consensus on a shared decision variable , minimizing a sum of subproblem objectives via the augmented Lagrangian to achieve convergence under convexity. The case study on portfolio optimization with historical prices and unstructured news demonstrates improved risk-adjusted returns and robust convergence across diverse LLMs, validating the synergy between precise optimization and contextual reasoning. The work offers a modular, privacy-preserving framework and prompts design guidance, with potential applicability to other decision-making domains requiring integration of structured and unstructured data.

Abstract

This paper introduces SOLID (Synergizing Optimization and Large Language Models for Intelligent Decision-Making), a novel framework that integrates mathematical optimization with the contextual capabilities of large language models (LLMs). SOLID facilitates iterative collaboration between optimization and LLMs agents through dual prices and deviation penalties. This interaction improves the quality of the decisions while maintaining modularity and data privacy. The framework retains theoretical convergence guarantees under convexity assumptions, providing insight into the design of LLMs prompt. To evaluate SOLID, we applied it to a stock portfolio investment case with historical prices and financial news as inputs. Empirical results demonstrate convergence under various scenarios and indicate improved annualized returns compared to a baseline optimizer-only method, validating the synergy of the two agents. SOLID offers a promising framework for advancing automated and intelligent decision-making across diverse domains.

Paper Structure

This paper contains 26 sections, 1 theorem, 5 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Framework of SOLID
  3. Theoretical Property and Considerations
  4. Case Study: Portfolio Optimization with Alternative Data
  5. Case Study: Portfolio Optimization with Alternative Data
  6. Experimental Setup
  7. Optimization Agent
  8. LLM Agent
  9. Results
  10. Portfolio Performance
  11. Agent Convergence
  12. Model Comparisons
  13. Conclusions
  14. Related Work
  15. Optimization Modeling:
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose Assumption assumption:1 holds. Then any sequence $\{(x^k,z^k,\lambda^k)\}$ produced by the ADMM iteration above is bounded (or has a convergent subsequence). Moreover, every limit point $(x^\star,z^\star,\lambda^\star)$ of this sequence satisfies: 1. Primal feasibility:$x^\star = z^\star$; 2

Figures (4)

  • Figure 1: Given a decision query from a user, our framework aims to perform optimal decision-making by leveraging the advantages of both the optimization model and the LLM model. We first illustrate the decision-making task by financial investment.
  • Figure 2: Panel A: the total portfolio value change by month; Panel B: Risk evaluation by month; Panel C: average stock weights under 5 strategies by month.
  • Figure 3: The coordination process of optimization and LLM agents in SOLID for exemplary stock - AMZN.
  • Figure 4: Comparisons of overall return rate and average risk by month between ChatGPT 4o-mini, 4o and o1-mini.

Theorems & Definitions (3)

  • Theorem 1: Convergence of ADMM
  • proof : Sketch of Proof
  • proof : Sketch of Proof