Solving General Natural-Language-Description Optimization Problems with Large Language Models

Optimization problems seek to find the best solution to an objective under a set of constraints, and have been widely investigated in real-world applications. Modeling and solving optimization problems in a specific domain typically require a combination of domain knowledge, mathematical skills, and programming ability, making it difficult for general users and even domain professionals. In this paper, we propose a novel framework called OptLLM that augments LLMs with external solvers. Specifically, OptLLM accepts user queries in natural language, convert them into mathematical formulations and programming codes, and calls the solvers to calculate the results for decision-making. In addition, OptLLM supports multi-round dialogues to gradually refine the modeling and solving of optimization problems. To illustrate the effectiveness of OptLLM, we provide tutorials on three typical optimization applications and conduct experiments on both prompt-based GPT models and a fine-tuned Qwen model using a large-scale selfdeveloped optimization dataset. Experimental results show that OptLLM works with various LLMs, and the fine-tuned model achieves an accuracy boost compared to the promptbased models. Some features of OptLLM framework have been available for trial since June 2023 (https://opt.alibabacloud.com/chat or https://opt.aliyun.com/chat).

Expressive Power of Graph Neural Networks for Quadratic Programs

Quadratic programming (QP) is the most widely applied category of problems in nonlinear programming. Many applications require real-time/fast solutions, though not necessarily with high precision. Existing methods either involve matrix decomposition or use the preconditioned conjugate gradient method. For relatively large instances, these methods cannot achieve the real-time requirement unless there is an effective precondition. Recently, graph neural networks (GNNs) opened new possibilities for QP. Some promising empirical studies of applying GNNs for QP tasks show that GNNs can capture key characteristics of an optimization instance and provide adaptive guidance accordingly to crucial configurations during the solving process, or directly provide an approximate solution. Despite notable empirical observations, theoretical foundations are still lacking. In this work, we investigate the expressive or representative power of GNNs, a crucial aspect of neural network theory, specifically in the context of QP tasks, with both continuous and mixed-integer settings. We prove the existence of message-passing GNNs that can reliably represent key properties of quadratic programs, including feasibility, optimal objective value, and optimal solution. Our theory is validated by numerical results.

ODE-based Learning to Optimize

Recent years have seen a growing interest in understanding acceleration methods through the lens of ordinary differential equations (ODEs). Despite the theoretical advancements, translating the rapid convergence observed in continuous-time models to discrete-time iterative methods poses significant challenges. In this paper, we present a comprehensive framework integrating the inertial systems with Hessian-driven damping equation (ISHD) and learning-based approaches for developing optimization methods through a deep synergy of theoretical insights. We first establish the convergence condition for ensuring the convergence of the solution trajectory of ISHD. Then, we show that provided the stability condition, another relaxed requirement on the coefficients of ISHD, the sequence generated through the explicit Euler discretization of ISHD converges, which gives a large family of practical optimization methods. In order to select the best optimization method in this family for certain problems, we introduce the stopping time, the time required for an optimization method derived from ISHD to achieve a predefined level of suboptimality. Then, we formulate a novel learning to optimize (L2O) problem aimed at minimizing the stopping time subject to the convergence and stability condition. To navigate this learning problem, we present an algorithm combining stochastic optimization and the penalty method (StoPM). The convergence of StoPM using the conservative gradient is proved. Empirical validation of our framework is conducted through extensive numerical experiments across a diverse set of optimization problems. These experiments showcase the superior performance of the learned optimization methods.

Learning to optimize: A tutorial for continuous and mixed-integer optimization

Learning to Optimize (L2O) stands at the intersection of traditional optimization and machine learning, utilizing the capabilities of machine learning to enhance conventional optimization techniques. As real-world optimization problems frequently share common structures, L2O provides a tool to exploit these structures for better or faster solutions. This tutorial dives deep into L2O techniques, introducing how to accelerate optimization algorithms, promptly estimate the solutions, or even reshape the optimization problem itself, making it more adaptive to real-world applications. By considering the prerequisites for successful applications of L2O and the structure of the optimization problems at hand, this tutorial provides a comprehensive guide for practitioners and researchers alike.

Bagging Improves Generalization Exponentially

Bagging is a popular ensemble technique to improve the accuracy of machine learning models. It hinges on the well-established rationale that, by repeatedly retraining on resampled data, the aggregated model exhibits lower variance and hence higher stability, especially for discontinuous base learners. In this paper, we provide a new perspective on bagging: By suitably aggregating the base learners at the parametrization instead of the output level, bagging improves generalization performances exponentially, a strength that is significantly more powerful than variance reduction. More precisely, we show that for general stochastic optimization problems that suffer from slowly (i.e., polynomially) decaying generalization errors, bagging can effectively reduce these errors to an exponential decay. Moreover, this power of bagging is agnostic to the solution schemes, including common empirical risk minimization, distributionally robust optimization, and various regularizations. We demonstrate how bagging can substantially improve generalization performances in a range of examples involving heavy-tailed data that suffer from intrinsically slow rates.

Revisiting Zeroth-Order Optimization for Memory-Efficient LLM Fine-Tuning: A Benchmark

In the evolving landscape of natural language processing (NLP), fine-tuning pre-trained Large Language Models (LLMs) with first-order (FO) optimizers like SGD and Adam has become standard. Yet, as LLMs grow {in size}, the substantial memory overhead from back-propagation (BP) for FO gradient computation presents a significant challenge. Addressing this issue is crucial, especially for applications like on-device training where memory efficiency is paramount. This paper proposes a shift towards BP-free, zeroth-order (ZO) optimization as a solution for reducing memory costs during LLM fine-tuning, building on the initial concept introduced by MeZO. Unlike traditional ZO-SGD methods, our work expands the exploration to a wider array of ZO optimization techniques, through a comprehensive, first-of-its-kind benchmarking study across five LLM families (Roberta, OPT, LLaMA, Vicuna, Mistral), three task complexities, and five fine-tuning schemes. Our study unveils previously overlooked optimization principles, highlighting the importance of task alignment, the role of the forward gradient method, and the balance between algorithm complexity and fine-tuning performance. We further introduce novel enhancements to ZO optimization, including block-wise descent, hybrid training, and gradient sparsity. Our study offers a promising direction for achieving further memory-efficient LLM fine-tuning. Codes to reproduce all our experiments are at https://github.com/ZO-Bench/ZO-LLM .

Rethinking the Capacity of Graph Neural Networks for Branching Strategy

Graph neural networks (GNNs) have been widely used to predict properties and heuristics of mixed-integer linear programs (MILPs) and hence accelerate MILP solvers. This paper investigates the capacity of GNNs to represent strong branching (SB) scores that provide an efficient strategy in the branch-and-bound algorithm. Although message-passing GNN (MP-GNN), as the simplest GNN structure, is frequently employed in the existing literature to learn SB scores, we prove a fundamental limitation in its expressive power -- there exist two MILP instances with different SB scores that cannot be distinguished by any MP-GNN, regardless of the number of parameters. In addition, we establish a universal approximation theorem for another GNN structure called the second-order folklore GNN (2-FGNN). We show that for any data distribution over MILPs, there always exists a 2-FGNN that can approximate the SB score with arbitrarily high accuracy and arbitrarily high probability. A small-scale numerical experiment is conducted to directly validate our theoretical findings.

DIG-MILP: a Deep Instance Generator for Mixed-Integer Linear Programming with Feasibility Guarantee

Mixed-integer linear programming (MILP) stands as a notable NP-hard problem pivotal to numerous crucial industrial applications. The development of effective algorithms, the tuning of solvers, and the training of machine learning models for MILP resolution all hinge on access to extensive, diverse, and representative data. Yet compared to the abundant naturally occurring data in image and text realms, MILP is markedly data deficient, underscoring the vital role of synthetic MILP generation. We present DIG-MILP, a deep generative framework based on variational auto-encoder (VAE), adept at extracting deep-level structural features from highly limited MILP data and producing instances that closely mirror the target data. Notably, by leveraging the MILP duality, DIG-MILP guarantees a correct and complete generation space as well as ensures the boundedness and feasibility of the generated instances. Our empirical study highlights the novelty and quality of the instances generated by DIG-MILP through two distinct downstream tasks: (S1) Data sharing, where solver solution times correlate highly positive between original and DIG-MILP-generated instances, allowing data sharing for solver tuning without publishing the original data; (S2) Data Augmentation, wherein the DIG-MILP-generated instances bolster the generalization performance of machine learning models tasked with resolving MILP problems.

A Human-on-the-Loop Optimization Autoformalism Approach for Sustainability

This paper outlines a natural conversational approach to solving personalized energy-related problems using large language models (LLMs). We focus on customizable optimization problems that necessitate repeated solving with slight variations in modeling and are user-specific, hence posing a challenge to devising a one-size-fits-all model. We put forward a strategy that augments an LLM with an optimization solver, enhancing its proficiency in understanding and responding to user specifications and preferences while providing nonlinear reasoning capabilities. Our approach pioneers the novel concept of human-guided optimization autoformalism, translating a natural language task specification automatically into an optimization instance. This enables LLMs to analyze, explain, and tackle a variety of instance-specific energy-related problems, pushing beyond the limits of current prompt-based techniques. Our research encompasses various commonplace tasks in the energy sector, from electric vehicle charging and Heating, Ventilation, and Air Conditioning (HVAC) control to long-term planning problems such as cost-benefit evaluations for installing rooftop solar photovoltaics (PVs) or heat pumps. This pilot study marks an essential stride towards the context-based formulation of optimization using LLMs, with the potential to democratize optimization processes. As a result, stakeholders are empowered to optimize their energy consumption, promoting sustainable energy practices customized to personal needs and preferences.

DSGD-CECA: Decentralized SGD with Communication-Optimal Exact Consensus Algorithm

Decentralized Stochastic Gradient Descent (SGD) is an emerging neural network training approach that enables multiple agents to train a model collaboratively and simultaneously. Rather than using a central parameter server to collect gradients from all the agents, each agent keeps a copy of the model parameters and communicates with a small number of other agents to exchange model updates. Their communication, governed by the communication topology and gossip weight matrices, facilitates the exchange of model updates. The state-of-the-art approach uses the dynamic one-peer exponential-2 topology, achieving faster training times and improved scalability than the ring, grid, torus, and hypercube topologies. However, this approach requires a power-of-2 number of agents, which is impractical at scale. In this paper, we remove this restriction and propose \underline{D}ecentralized \underline{SGD} with \underline{C}ommunication-optimal \underline{E}xact \underline{C}onsensus \underline{A}lgorithm (DSGD-CECA), which works for any number of agents while still achieving state-of-the-art properties. In particular, DSGD-CECA incurs a unit per-iteration communication overhead and an $\tilde{O}(n^3)$ transient iteration complexity. Our proof is based on newly discovered properties of gossip weight matrices and a novel approach to combine them with DSGD's convergence analysis. Numerical experiments show the efficiency of DSGD-CECA.