Mixed Integer Goal Programming for Personalized Meal Optimization (arXiv 2505.12345)

Forum topic · 小凯 · 2026-05-18

Summary

This post summarizes a machine learning / operations research paper by Francisco Aguilera Moreno (arXiv:2505.12345) proposing Mixed Integer Goal Programming (MIGP) for personalized meal optimization. Traditional diet optimization suffers from two flaws: continuous variables yield impractical fractional servings (1.7 eggs, 0.37 bananas), and hard nutrient constraints become infeasible when targets conflict. A systematic review of 56 diet optimization papers found none combining integer programming with goal programming. MIGP uses integer variables for realistic serving counts and goal programming deviations for soft nutrient targets, balanced via inverse-target normalization. The authors show goal-programming deviations structurally buffer integrality gaps, and for meals with 15+ foods, integer solutions match the continuous optimum in all benchmark instances. Across 810 instances (30 USDA foods, 9 configurations, 3 methods), MIGP found strictly better solutions than post-hoc rounding in 66% of cases (never worse) with 100% feasibility, versus 48% for hard-constraint integer programming. Using the open-source HiGHS solver, typical meal-scale solve times stay under 100 ms. The implementation ships as an open-source Python module integrated into an interactive meal planning app.

Paper Overview

Field: ML Author: Francisco Aguilera Moreno Published: 2026-05-17 arXiv: 2505.12345

Abstract (Translated)

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 authors 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. The paper describes the integrality gap in the goal programming context and identifies a deviation-absorption property: goal programming deviation variables buffer the cost of requiring integer servings, making the gap structurally smaller than in hard-constraint mixed integer programming. For meals containing more than 15 foods, integer solutions match the continuous optimum in every benchmark instance. A computational evaluation spanning 810 instances (30 USDA foods, 9 configurations, 3 methods) shows MIGP finds goal programming solutions strictly better than post-hoc rounding in 66% of cases (never worse), while maintaining 100% feasibility; hard-constraint integer programming reaches only 48%. Using the open-source HiGHS solver, typical meal-scale solve times remain under 100 milliseconds. The implementation is available as an open-source Python module integrated into an interactive meal planning application.

Key Takeaways

MIGP combines integer programming (practical whole servings) with goal programming (soft nutrient targets) for the first time in diet optimization, per a review of 56 prior papers.
Goal programming deviation variables absorb the cost of integrality, shrinking the integrality gap relative to hard-constraint MIP.
Empirical results: 810 instances, MIGP strictly better than post-hoc rounding in 66% of cases, 100% feasibility, solve times < 100 ms with HiGHS.
Open-source Python implementation integrated into an interactive meal planning app.

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