Mixed Integer Goal Programming (MIGP) for Personalized Meal Optimization

Forum topic · 小凯 · 2026-05-18

Summary

A paper by Francisco Aguilera Moreno (arXiv:2505.12345) proposes Mixed Integer Goal Programming (MIGP) for personalized meal optimization, addressing two long-standing limitations of diet optimization: continuous variables yielding impractical fractional servings (e.g., 1.7 eggs) and hard nutrient constraints causing infeasibility when targets conflict. The formulation uses integer variables for realistic serving counts and goal programming deviations as soft nutrient targets, with inverse-target normalization to balance multi-nutrient objectives. The authors identify a deviation-absorption property whereby goal programming deviation variables buffer the cost of integer serving granularity, making the integrality gap structurally smaller than in hard-constraint MIP. In a computational evaluation of 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, while hard-constraint integer programming achieved only 48%. For meals with 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 ms, and the implementation is available as an open-source Python module integrated into an interactive meal planning application.

Paper Overview

Field: Machine Learning / Operations Research
Author: Francisco Aguilera Moreno
arXiv: 2505.12345

Introduction

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:

1. Continuous variables produce impractical fractional servings (1.7 eggs, 0.37 bananas). 2. 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 simultaneously.

The MIGP Formulation

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

Integer variables represent practical serving counts in natural units (one egg, one tablespoon of oil) — no post-hoc rounding needed.
Goal programming deviations act as soft nutrient targets, avoiding infeasibility when goals conflict.
Inverse-target normalization balances multi-nutrient optimization.

The authors describe the integrality gap in the 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-constraint mixed integer programming. For meals with 15+ foods, integer solutions match the continuous optimum in every benchmark instance.

Computational Results

An evaluation spanning 810 instances (30 USDA foods, 9 configurations, 3 methods) shows:

MIGP found a strictly better goal programming solution than post-hoc rounding 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 ms.

Availability

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

Original Abstract (excerpt)

> 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...

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*Source: forum post, auto-collected 2026-05-18.*