Mixed Integer Goal Programming (MIGP) for Personalized Meal Optimization

Forum topic · 小凯 · 2026-05-18

Summary

A new paper by Francisco Aguilera Moreno (arXiv:2505.12345) proposes Mixed Integer Goal Programming (MIGP) for personalized meal optimization, addressing two persistent limitations of classical diet optimization: continuous variables yield impractical fractional servings (e.g., 1.7 eggs), and hard nutrient constraints become infeasible when targets conflict. The formulation uses integer variables for practical serving counts, goal programming deviation variables for soft nutrient targets, and inverse-target normalization to balance multi-nutrient optimization. The authors identify a deviation-absorption property whereby goal programming deviations buffer the cost of integer servings, structurally reducing the integrality gap. In a computational evaluation across 810 instances (30 USDA foods, 9 configurations, 3 methods), MIGP found strictly better solutions than post-hoc rounding of goal programming in 66% of cases (never worse) with 100% feasibility, versus 48% for hard-constraint integer programming. For meals with more than 15 foods, integer solutions matched the continuous optimum in every benchmark instance. Using the open-source HiGHS solver, typical meal-scale solve times stay under 100 milliseconds. The implementation is available as an open-source Python module integrated into an interactive meal planning application.

Overview

Field: Machine Learning / Operations Research
Author: Francisco Aguilera Moreno
Published: 2026-05-17
arXiv: 2505.12345

Key Points

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.

The paper proposes Mixed Integer Goal Programming (MIGP) for personalized meal optimization:

Integer variables represent practical serving counts, so portions come in natural units (one egg, one tablespoon of oil) without post-hoc rounding.
Goal programming deviations act as soft nutrient targets, allowing feasible solutions even when nutrient goals conflict.
Inverse-target normalization balances multi-nutrient optimization.
The authors characterize the integrality gap in a goal programming context and identify a deviation-absorption property: deviation variables buffer the cost of requiring integer servings, making the gap structurally smaller than in hard-constrained mixed integer programming. For meals with more than 15 foods, integer solutions matched the continuous optimum in every benchmark instance.

Results

Computational evaluation across 810 instances (30 USDA foods, 9 configurations, 3 methods).
MIGP found strictly better solutions than post-hoc rounding of a goal programming solution in 66% of cases (never worse), while maintaining 100% feasibility; hard-constraint integer programming achieved only 48%.
Using the open-source HiGHS solver, typical meal-scale solve times remain under 100 milliseconds.

Availability

The implementation is released as an open-source Python module integrated into an interactive meal planning application.

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