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

🕒 Published (v1): 2025-11-19 07:44 UTC · Source: Arxiv · Venue: NEURIPS 2025 · link

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TL;DR

SOLID couples a mathematical optimization solver with an LLM agent via an ADMM-inspired iterative coordination loop, letting each agent iteratively propose decisions that are reconciled by a coordinator using dual prices and deviation penalties. Applied to portfolio optimization over 60 NASDAQ stocks throughout 2024, SOLID outperforms both pure-optimizer and pure-LLM baselines on annualized returns while maintaining intermediate risk. The framework retains convergence guarantees under convexity assumptions and is modular enough to preserve data privacy between agents.

Problem

Mathematical optimization models handle structured numerical inputs well but cannot directly consume unstructured contextual data (news, narratives). LLMs can reason over unstructured text but are unreliable for constrained numerical optimization. No prior framework enables principled, iterative, theoretically-grounded coordination between these two complementary decision-making paradigms.

Method

SOLID casts the joint optimization-LLM decision problem as a consensus optimization (Eq. 5: min f(x)+g(z) s.t. x=z) and applies ADMM-style decomposition. At each iteration: (1) the optimization agent (Markowitz mean-variance) maximizes its augmented utility penalized for deviation from the current public plan; (2) the LLM agent receives the public plan and its dual price λ_llm via a Chain-of-Thought prompt, responds with discrete confidence levels ("Very High"→0.6, …, "Very Low"→0), which are mapped to continuous weights; (3) a coordinator updates dual prices and reconciles the two agents' proposals into a new public plan. The LLM temperature is fixed at 0 to reduce stochasticity. News is sourced via Perplexity.ai; prices via Yahoo Finance API.

Key Contributions

  • Novel ADMM-inspired coordination mechanism for LLM and optimization agents via dual prices and quadratic deviation penalties.
  • Convergence proof under standard convexity assumptions (proper, closed, convex f and g; feasibility; nonempty minimizer set); analysis of failure modes when LLM decisions introduce non-convexity.
  • Demonstration that communicating dual prices to the LLM as "economic costs" (a semantic abstraction) is an effective prompting strategy for inducing feasible, consensus-aligned decisions.
  • Modular, privacy-preserving design: agents operate on separate datasets and communicate only through the public plan and price signals.

Results

  • Over 12 monthly rebalancing periods (full year 2024), SOLID variants (LLM+OPT, LLM_sparse+OPT) achieved the highest final portfolio values starting from $10,000, outperforming pure OPT, pure LLM, and simple averaging (AVG).
  • Pure OPT exhibited pronounced volatility spikes around months 4–5; SOLID achieved intermediate risk (lower than OPT, higher than LLM_sparse).
  • LLM_sparse+OPT with GPT-4o achieved "exceptionally high returns" (specific return figures not reported in the text).
  • Results held across ChatGPT-4o-mini, 4o, and o1-mini, validating robustness to LLM choice.
  • Exact convergence observed at some months (e.g., months 1 and 7 for AMZN); progressive alignment observed at others.

Limitations

  • Convergence guarantees hold only under convexity; LLM discrete outputs introduce non-convexity that can prevent convergence.
  • Backtesting covers only 60 NASDAQ stocks over one year (2024); generalizability to other asset classes or time horizons is unvalidated.
  • No absolute return figures are provided; comparisons are qualitative (portfolio value curves) without statistical significance testing.
  • The LLM prompt is highly engineered and includes an explicit instruction that the optimizer "has consistently outperformed by 10–20%", which biases the LLM toward the optimizer's proposal and may not transfer to domains where such a prior is unjustified.
  • News retrieval via Perplexity.ai introduces an external dependency whose summarization quality is uncontrolled.
  • The semantic discretization of LLM outputs (7 confidence levels) loses information and may not generalize beyond ordinal allocation tasks.

Relevance to Agentic AI / LLM Agents

SOLID is a concrete instantiation of multi-agent coordination where an LLM agent and a classical solver agent iteratively negotiate a shared decision under a principled economic signaling protocol—directly relevant to research on structured LLM-tool collaboration and hybrid agentic systems. The use of dual prices as inter-agent communication tokens is a novel prompt-engineering insight: economic concepts serve as interpretable signals that steer LLM behavior without requiring the model to perform numerical optimization directly. This bridges the gap between symbolic/formal planning agents and language model agents, a central challenge in building reliable agentic AI for real-world decision tasks. The modular design and privacy-preserving property also suggest a path toward multi-organization agentic workflows where agents cannot share raw data.