Abstract
Determining what to eat to satisfy nutritional requirements is one of the oldest optimization problems in operations research, yet existing formulations have two persistent limitations: continuous variables produce impractical fractional servings (1.7 eggs, 0.37 bananas), and hard nutrient constraints cause infeasibility when targets conflict. A systematic review of 56 diet optimization papers found that none combine integer programming with goal programming to address both issues. We propose Mixed Integer Goal Programming (MIGP) for personalized meal optimization. The formulation uses integer variables for practical serving counts and goal programming deviations for soft nutrient targets, with inverse-target normalization to balance multi-nutrient optimization. Per-food serving granularity allows natural units (one egg, one tablespoon of oil) without post-hoc rounding. We characterize the integrality gap in the goal programming context and identify a deviation absorption property: GP deviation variables buffer the cost of requiring integer servings, making the gap structurally smaller than in hard-constraint MIP. For meals with 15+ foods, the integer solution matches the continuous optimum in every benchmark instance. A computational evaluation across 810 instances (30 USDA foods, 9 configurations, 3 methods) shows MIGP finds strictly better solutions than GP with post-hoc rounding in 66% of cases (never worse) while maintaining 100% feasibility; hard-constraint IP achieves only 48%. Solve times stay under 100 ms for typical meal sizes using the open-source HiGHS solver. The implementation is available as an open-source Python module integrated into an interactive meal planning application.
More in this topicMixed-integer linear programming (MILP) optimizes linear objectives and constraints when some decisions must be integer-valued.
See all 50 papers → Oct 6, 2026
Dietary nutrition planning plays an important role in chronic disease management and maintaining a healthy body. In applications, it must simultaneously satisfy personalized constraints and reasonable multidimensional nutritional goals. These two aspects often conflict, and user constraints evolve with feedback, resulting in a substantial gap between generic guidelines and executable plans. To bridge this gap, we first propose the personalized fully quantified multiobjective dietary planning problem (MDP). To tackle MDP, we develop a nutrition agent, ShanLiangRen. The system first transforms dietary specifications, nutrient data, user attributes and natural language requirements into an individualized constrained planning instance. It then employs an exact retrieval-augmented generation method to shrink the feasible candidate set from a large scale ingredient and recipe space. Finally, it adopts a refinement guided by Pareto principles, where an LLM iteratively revises candidate plans under deterministic nutrition computation and feedback from constraint verification. The system outputs fully quantified meal plans with explicit ingredients and portion sizes, together with reports on nutrition compliance that show constraint satisfaction and nutrient interval attainment. We have released the system online as a WeChat Program, ShanLiangRen. A demo video is available at https://www.youtube.com/watch?v=652OtY5VlGA.
Miao Xie, Xiao Zhang, Yuan Wang +2
College of Information and Electrical Engineering, China Agricultural University · Key Laboratory of Precision Nutrition and Food Quality, China Agricultural University
Jun 22, 2026
Configuring the hyperparameters of Mixed-integer programming (MIP) solvers is a high-dimensional, instance-dependent optimization problem where suboptimal settings can degrade solving time by orders of magnitude. Default configurations are often suboptimal, while traditional tuning methods either suffer from the ``cold-start'' problem and inefficient search or heavily rely on expert experience. This paper introduces GRIMIP (General Reasoning for Instance-specific MIP configuration), a novel hybrid intelligence framework that synergistically integrates the semantic reasoning capabilities of Large Language Models (LLMs) with the sample-efficient search of Bayesian Optimization (BO). GRIMIP enables the LLM to function as a complete probabilistic surrogate within the BO loop, significantly improving performance and reducing sampling and evaluation costs. On seven benchmarks including MIPLIB, GRIMIP achieves over 40% reduction in Primal-Dual Integral on hard instances, outperforming SMAC and other LLM-assisted BO methods. By granting LLMs sufficient autonomy, GRIMIP combines the expert-level reasoning of LLMs with the efficient search of BO, achieving state-of-the-art performance.
Yidong Luo, Xuemin Chen, Chenguang Wang +3
School of Data Science, The Chinese University of Hong Kong, Shenzhen · School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen · Noah’s Ark Lab, Huawei
Oct 6, 2025
Data-driven inverse optimization for mixed-integer linear programs (MILPs), which seeks to learn an objective function and constraints consistent with observed decisions, is important for building accurate mathematical models in a variety of domains, including power systems and scheduling. However, to the best of our knowledge, existing data-driven inverse optimization methods primarily focus on learning objective functions under known constraints, and learning both objective functions and constraints from data for MILPs remains largely unexplored. In this paper, we propose a two-stage approach for a class of inverse optimization problems in which the objective is a linear combination of given feature functions and the constraints are parameterized by unknown functions and thresholds. Our method first learns the constraints and then, conditioned on the learned constraints, estimates the objective-function weights. On the theoretical side, we provide finite-sample guarantees for solving the proposed inverse optimization problem. To this end, we develop statistical learning tools for pseudo-metric spaces under sub-Gaussian assumptions and use them to derive a learning-theoretic framework for inverse optimization with both unknown objectives and constraints. On the experimental side, we demonstrate that our method successfully solves inverse optimization problems on scheduling instances formulated as ILPs with up to 100 decision variables.
Akira Kitaoka