Eating 1.7 Eggs and 0.37 Bananas? Mixed Integer Goal Programming Fixes Unfeasible Diet-App Recommendations

Forum topic · 小凯 · 2026-05-17

Summary

A forum post discusses a 2026 arXiv paper (arXiv:2605.13849) by Francisco Aguilera Moreno proposing Mixed Integer Goal Programming (MIGP) for personalized meal optimization. Classic diet optimizers output impractical fractional servings like 1.7 eggs or 0.37 bananas, and hard nutritional constraints often produce infeasible solutions when they conflict. MIGP combines integer programming (whole-number servings) with goal programming (soft targets whose deviation variables absorb the cost of integer rounding). In tests across 810 instances using 30 USDA foods, 9 configurations, and 3 methods, MIGP achieved 100% feasibility, outperformed the naive 'continuous solve plus rounding' approach in 66% of cases (never worse), and solved in under 100 ms with the open-source HiGHS solver, versus only 48% feasibility for hard-constraint integer programming. Notably, with 15+ foods per meal, integer solutions matched the continuous optimum exactly. The post also offers critical caveats: nutritional realism beyond math, the 15-food condition, and preference modeling remain open questions.

Eating 1.7 Eggs and 0.37 Bananas? MIGP Fixes Unfeasible Diet Recommendations

A review of: Mixed Integer Goal Programming for Personalized Meal Optimization with User-Defined Serving Granularity — Francisco Aguilera Moreno (arXiv:2605.13849, cs.AI, March 12, 2026)

| Item | Details | |------|---------| | Paper | Mixed Integer Goal Programming for Personalized Meal Optimization with User-Defined Serving Granularity | | Author | Francisco Aguilera Moreno (single author) | | arXiv | 2605.13849 (cs.AI) | | Core contribution | Integer programming + goal programming to fix fractional servings and infeasible hard constraints; 810 test instances, beats rounding in 66% of cases | | Link | https://arxiv.org/abs/2605.13849 |

Your diet app tells you to eat 1.7 eggs and 0.37 bananas. Do you slice 0.37 of a banana? Beat 0.7 of an egg? Most people would just give up.

1. Two Flaws of Classical Nutritional Optimization

Nutritional optimization is old news — it shows up in gyms, hospitals, and military canteens. The task: pick a combination of foods so that protein, carbs, fats, vitamins, etc. all fall within target ranges. But classical mathematical methods have two inherent flaws:

Flaw one: fractional servings. Variables are continuous, so the "optimal" plan is 1.7 eggs plus 0.37 bananas. Mathematically perfect, practically nonsense. The usual patch is rounding after the fact — but rounding 0.37 bananas to 0 leaves a protein gap, and rounding 1.7 eggs to 2 overshoots cholesterol.

Flaw two: infeasible hard constraints. Set "protein ≥ 60g," "fat ≤ 60g," "carbs ≤ 250g," "vitamin C ≥ 100mg" — foods meeting the protein target are often high in fat. When hard constraints conflict, the optimizer reports: no feasible solution. Your needs probably aren't impossible — your constraints are just too rigid.

2. MIGP: Turning "Must" into "Try"

The paper's move: convert hard constraints into soft goals while keeping integer servings.

Integer Programming: variables must be whole numbers. Eggs are 0, 1, 2... never 1.7.
Goal Programming: constraints become "get as close as possible," with deviation penalties in the objective function.

The key insight is deviation absorption: the deviation variables already present in goal programming naturally absorb the nutritional cost of integer rounding — no extra penalties or corrections needed. MIGP doesn't patch a continuous optimum; it searches directly in integer space while goal programming absorbs the discretization cost.

Remarkably, when a meal includes 15+ foods, the integer solution matches the continuous optimum exactly on all benchmark instances. When your food variety is rich enough, an integer combination naturally achieves the continuous-optimal nutrition profile. Diversity is good for optimization, not just health.

3. Results

810 instances, 30 USDA-standard foods, 9 configurations, 3 methods:

MIGP: 100% feasibility; better than GP + rounding in 66% of cases (never worse)
Hard-constraint IP: feasible in only 48% of cases — more than half the time you get no plan at all

Solve times are under 100 ms at typical meal sizes, using the open-source HiGHS solver and an open-source Python module. You can run it on your laptop.

4. Honest Caveats

Nutritional accuracy is beyond the math. Same calories and protein — fried chicken and chicken breast affect the body differently. Fiber types, fatty-acid ratios, micronutrient interactions: how much of this can mathematical optimization cover? Unclear.

The 15-food condition. For a single meal, 15 foods is a lot. Could the condition encourage "eating more for the optimizer's sake"?

Preference modeling. User-defined serving granularity (eggs by unit, oil by tablespoon) is one thing, but modeling taste, allergies, and dislikes is a discipline of its own. The paper doesn't go deep here.

5. Verdict

A systematic review of 56 papers found that not one simultaneously addressed fractional servings and hard-constraint infeasibility. This is the classic "you assume a solution exists, but it doesn't" situation.

The post recalls a WWII anecdote: optimization algorithms assigned soldiers rations heavy in liver — because liver is rich in nearly every nutrient. Theoretically perfect; no soldier wanted to eat liver daily. Math finds the optimal solution; reality needs a feasible one. The gap between them is what this paper sets out to fill.

Whether MIGP ends up in real diet apps is unknown — but next time your phone says "eat 1.7 eggs," you can say: "No, I'll eat 2. Recompute."

References

1. Aguilera Moreno, F. (2026). Mixed Integer Goal Programming for Personalized Meal Optimization. arXiv:2605.13849. 2. Charnes, A., et al. (1955). Optimal Estimation of Executive Compensation by Linear Programming. Management Science. 3. Romero, C. (1991). Handbook of Critical Issues in Goal Programming. Pergamon Press. 4. USDA FoodData Central. https://fdc.nal.usda.gov/