Mixed Integer Goal Programming for Personalized Meal Optimization with User-Defined Serving Granularity

Francisco Aguilera Moreno

cs.AIMar 12, 2026

Mixed Integer Goal Programming for Personalized Meal Optimization with User-Defined Serving Granularity

Authors: Francisco Aguilera Moreno

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.

Figures & tables

Approach	Variables	Constraints	Feasible	Servings	Optimal
Classical LP	Continuous	Hard	No	Fractional	Yes
GP (Gerdessen)	Continuous	Soft	Yes	Fractional	Yes
IP (Benvenuti)	Binary	Hard	No	Binary	Yes
AI/ML	—	Soft	Yes	Varies	No
MIGP (ours)	Integer	Soft	Yes	Integer	Yes

Table 1: Comparison of diet optimization approaches. MIGP uniquely combines integer serving variables with soft goal constraints, achieving both feasibility and optimality with practical serving outputs.

Sets
Symbol	Description
$\mathcal{F}=\{1,\ldots,n\}$	Selected foods
$\mathcal{M}=\{\text{cal, prot, carbs, fat}\}$	Macronutrients
Parameters
$a_{i,m}$	Nutrient $m$ per 100 g of food $i$ (kcal or grams)
$s_{i}$	Serving size in grams for food $i$

Table 2: Notation used in the MIGP formulation.

Figure 1: Integrality cost vs. number of foods. (a) Absolute integrality cost (integer minus continuous objective) across all instances. (b) Percentage gap, computed only for instances where the LP has positive deviation (annotation shows subsample size). Shaded regions show interquartile ranges; dots indicate medians. Positive gaps occur only with 8 foods; 15 and 25 food instances always achieve zero integrality cost.

Figure 2: MIGP objective vs. GP+rounding objective for all 270 instances. Points below the diagonal (dashed) indicate MIGP superiority. MIGP finds a strictly better solution in 66% of instances and is never worse. The cluster near the diagonal at high objective values corresponds to ambitious configurations where both methods struggle equally.

Config	Foods	Serving range	Calories	Macro split
Small	8	Loose: $[0,10]$	600	30/45/25
Medium	15		800	30/45/25
Large	25		1000	30/45/25
Small	8	Tight: $[0,4]$	600	35/40/25
Medium	15		800	35/40/25
Large	25		1000	35/40/25

Table 3: Benchmark configurations. Macro split is protein/carbs/fat percentage. Ambitious configs force a minimum of 1 serving per food, creating challenging targets.

Method	Feasibility	Median Obj.	Median Max Dev.	Macros $\leq$ 5%	Median Time
MIGP	100%	0.1410	6.3%	75%	19.9
GP+Round	100%	0.5286	21.6%	0%	1.0
Hard-IP	48%	0.0911	4.0%	100%	3.5

Table 4: Comparison of optimization methods across all benchmark configurations.

Figure 3: Feasibility rates by constraint tightness (Y-axis capped at 105% for readability). MIGP maintains 100% feasibility regardless of configuration. Hard-Constraint IP degrades as targets become more restrictive, failing entirely on ambitious instances where every food must contribute at least 1 serving. GP+Rounding (not shown) also achieves 100% feasibility since rounding preserves serving bounds.

Figure 4: Median solve time vs. number of foods (log scale). MIGP scales from 13 ms to 1.1 s as problem size increases. GP+Rounding and Hard-IP remain under 5 ms. All methods are fast enough for interactive use at typical meal sizes (8–15 foods).

Figure 5: Maximum macro deviation by problem size and method. Hard-IP results shown only for feasible instances (

N

shown below each box); the 140 infeasible instances (51.9%) are excluded, creating a strong selection bias toward easy cases. Hard-IP’s apparent low deviations reflect its

\pm

5% tolerance band, not superior optimization. Wide ranges for MIGP and GP+Rounding are driven by ambitious configurations where targets are inherently unachievable.

Macro	% split	kcal/g	Calculation	Target
Calories	—	—	$600$	600.0 kcal
Protein	30%	4	$600\times 0.30\;/\;4=45.0$	45.0 g
Carbs	45%	4	$600\times 0.45\;/\;4=67.5$	67.5 g
Fat	25%	9	$600\times 0.25\;/\;9=16.7$	16.7 g

Table 5: Example A: target derivation from 600 kcal with 30/45/25 (P/C/F%).

			Per 100 g				Per serving
Food	Srv (g)	Range	kcal	P	C	F	kcal	P	C	F
Chicken breast	50	[0, 10]	165	31.0	0.0	3.6	82.5	15.5	0.0	1.8
White rice	50	[0, 10]	130	2.7	28.2	0.3	65.0	1.4	14.1	0.1
Broccoli	50	[0, 8]	35	2.4	7.2	0.4	17.5	1.2	3.6	0.2
Avocado	30	[0, 6]	160	2.0	8.5	14.7	48.0	0.6	2.5	4.4
Olive oil	15	[0, 4]	884	0.0	0.0	100.0	132.6	0.0	0.0	15.0

Table 6: Example A: food inputs. Target: 600 kcal, 30/45/25 (P/C/F%). Derived gram targets: 45.0g protein, 67.5g carbs, 16.7g fat.

Food (serving)	Qty	kcal	P (g)	C (g)	F (g)
Chicken breast (50 g)	2	165.0	31.0	0.0	3.6
White rice (50 g)	3	195.0	4.0	42.3	0.5
Broccoli (50 g)	5	87.5	6.0	18.0	1.0
Avocado (30 g)	3	144.0	1.8	7.7	13.2
Total		591.5	42.9	68.0	18.3
Target		600.0	45.0	67.5	16.7

Table 7: Example A: MIGP solution breakdown. Each row shows one food’s contribution; the deviation row shows (actual

-

target) / target as a percentage.

	LP Continuous	GP+Round	MIGP
Chicken breast	2.33	2	2
White rice	3.88	4	3
Broccoli	1.71	2	5
Avocado	2.62	3	3
kcal	600.0 (+0.0%)	604.0 (+0.7%)	591.5 (-1.4%)
P (g)	45.0 (+0.0%)	40.6 (-9.8%)	42.9 (-4.8%)

Table 8: Example A: side-by-side method allocations. LP Continuous shows fractional servings (lower bound

z_{LP}

); GP+Round rounds each food to the nearest integer; MIGP finds the globally optimal integer solution.

Method	Feasible	Obj.	Max dev.	Cal	Prot	Carbs	Fat
MIGP	Yes	0.1654	+9.7%	-1.4	-4.8	+0.7	+9.7
GP+Round	Yes	0.2298	9.8%	0.7	9.8	5.6	7.0
Hard-IP	No	—	—	—	—	—	—

Table 9: Example A: method comparison with per-macro deviations (%).

Macro	% split	kcal/g	Calculation	Target
Calories	—	—	$800$	800.0 kcal
Protein	35%	4	$800\times 0.35\;/\;4=70.0$	70.0 g
Carbs	40%	4	$800\times 0.40\;/\;4=80.0$	80.0 g
Fat	25%	9	$800\times 0.25\;/\;9=22.2$	22.2 g

Table 10: Example B: target derivation from 800 kcal with 35/40/25 (P/C/F%).

			Per 100 g				Per serving
Food	Srv (g)	Range	kcal	P	C	F	kcal	P	C	F
Chicken breast	50	[0, 8]	165	31.0	0.0	3.6	82.5	15.5	0.0	1.8
Salmon fillet	50	[0, 8]	208	20.4	0.0	13.4	104.0	10.2	0.0	6.7
White rice	50	[0, 8]	130	2.7	28.2	0.3	65.0	1.4	14.1	0.1
Sweet potato	50	[0, 8]	90	2.0	20.7	0.1	45.0	1.0	10.3	0.1
Broccoli	50	[0, 8]	35	2.4	7.2	0.4	17.5	1.2	3.6	0.2

Table 11: Example B: food inputs. Target: 800 kcal, 35/40/25 (P/C/F%). Derived gram targets: 70.0g protein, 80.0g carbs, 22.2g fat.

Food (serving)	Qty	kcal	P (g)	C (g)	F (g)
Chicken breast (50 g)	4	330.0	62.0	0.0	7.2
White rice (50 g)	2	130.0	2.7	28.2	0.3
Sweet potato (50 g)	5	225.0	5.0	51.8	0.2
Olive oil (15 g)	1	132.6	0.0	0.0	15.0
Total		817.6	69.7	80.0	22.8
Target		800.0	70.0	80.0	22.2

Table 12: Example B: MIGP solution breakdown. Each row shows one food’s contribution; the deviation row shows (actual

-

target) / target as a percentage.

	LP Continuous	GP+Round	MIGP
Chicken breast	2.55	3	4
White rice	0.00	0	2
Sweet potato	6.26	6	5
Broccoli	3.76	4	0
Olive oil	0.00	0	1
Whole eggs	3.13	3	0

Table 13: Example B: side-by-side method allocations. LP Continuous shows fractional servings (lower bound

z_{LP}

); GP+Round rounds each food to the nearest integer; MIGP finds the globally optimal integer solution.

Method	Feasible	Obj.	Max dev.	Cal	Prot	Carbs	Fat
MIGP	Yes	0.0507	+2.4%	+2.2	-0.4	-0.1	+2.4
GP+Round	Yes	0.1447	8.9%	2.5	8.9	2.3	0.8
Hard-IP	Yes	0.0899	4.1%	4.1	0.4	1.1	3.5

Table 14: Example B: method comparison with per-macro deviations (%).

Macro	% split	kcal/g	Calculation	Target
Calories	—	—	$600$	600.0 kcal
Protein	40%	4	$600\times 0.40\;/\;4=60.0$	60.0 g
Carbs	35%	4	$600\times 0.35\;/\;4=52.5$	52.5 g
Fat	25%	9	$600\times 0.25\;/\;9=16.7$	16.7 g

Table 15: Example C: target derivation from 600 kcal with 40/35/25 (P/C/F%).

			Per 100 g				Per serving
Food	Srv (g)	Range	kcal	P	C	F	kcal	P	C	F
Chicken breast	50	[1, 6]	165	31.0	0.0	3.6	82.5	15.5	0.0	1.8
Salmon fillet	50	[1, 6]	208	20.4	0.0	13.4	104.0	10.2	0.0	6.7
White rice	50	[1, 6]	130	2.7	28.2	0.3	65.0	1.4	14.1	0.1
Quinoa	50	[1, 6]	120	4.4	21.3	1.9	60.0	2.2	10.7	0.9
Avocado	30	[1, 6]	160	2.0	8.5	14.7	48.0	0.6	2.5	4.4

Table 16: Example C: food inputs. Target: 600 kcal, 40/35/25 (P/C/F%). Derived gram targets: 60.0g protein, 52.5g carbs, 16.7g fat.

Food (serving)	Qty	kcal	P (g)	C (g)	F (g)
Chicken breast (50 g)	2	165.0	31.0	0.0	3.6
Salmon fillet (50 g)	1	104.0	10.2	0.0	6.7
White rice (50 g)	2	130.0	2.7	28.2	0.3
Quinoa (50 g)	1	60.0	2.2	10.7	0.9
Avocado (30 g)	1	48.0	0.6	2.5	4.4
Olive oil (15 g)	1	132.6	0.0	0.0	15.0

Table 17: Example C: MIGP solution breakdown. Each row shows one food’s contribution; the deviation row shows (actual

-

target) / target as a percentage.

	LP Continuous	GP+Round	MIGP
Chicken breast	2.06	2	2
Salmon fillet	1.00	1	1
White rice	1.22	1	2
Quinoa	1.00	1	1
Avocado	1.00	1	1
Olive oil	1.00	1	1

Table 18: Example C: side-by-side method allocations. LP Continuous shows fractional servings (lower bound

z_{LP}

); GP+Round rounds each food to the nearest integer; MIGP finds the globally optimal integer solution.

Method	Feasible	Obj.	Max dev.	Cal	Prot	Carbs	Fat
MIGP	Yes	1.5557	+121.2%	+28.3	-5.7	+0.5	+121.2
GP+Round	Yes	1.5777	123.9%	26.2	1.9	5.8	123.9
Hard-IP	No	—	—	—	—	—	—

Table 19: Example C: method comparison with per-macro deviations (%).

Macro	% split	kcal/g	Calculation	Target
Calories	—	—	$1000$	1000.0 kcal
Protein	10%	4	$1000\times 0.10\;/\;4=25.0$	25.0 g
Carbs	75%	4	$1000\times 0.75\;/\;4=187.5$	187.5 g
Fat	15%	9	$1000\times 0.15\;/\;9=16.7$	16.7 g

Table 20: Example D: target derivation from 1000 kcal with 10/75/15 (P/C/F%).

			Per 100 g				Per serving
Food	Srv (g)	Range	kcal	P	C	F	kcal	P	C	F
Oats	40	[0, 8]	389	16.9	66.3	6.9	155.6	6.8	26.5	2.8
Honey	15	[0, 10]	304	0.3	82.4	0.0	45.6	0.0	12.4	0.0
Banana	120	[0, 3]	89	1.1	22.8	0.3	106.8	1.3	27.4	0.4
Dates	30	[0, 8]	282	2.5	75.0	0.4	84.6	0.8	22.5	0.1
Peanut butter	32	[0, 4]	588	25.1	19.6	50.4	188.2	8.0	6.3	16.1

Table 21: Example D: food inputs. Target: 1000 kcal, 10/75/15 (P/C/F%). Derived gram targets: 25.0g protein, 187.5g carbs, 16.7g fat.

Food (serving)	Qty	kcal	P (g)	C (g)	F (g)
Oats (40 g)	3	466.8	20.3	79.6	8.3
Honey (15 g)	1	45.6	0.0	12.4	0.0
Banana (120 g)	3	320.4	4.0	82.1	1.1
Dark chocolate chips (20 g)	1	109.2	1.1	12.1	6.3
Total		942.0	25.4	186.1	15.6
Target		1000.0	25.0	187.5	16.7

Table 22: Example D: MIGP solution breakdown. Each row shows one food’s contribution; the deviation row shows (actual

-

target) / target as a percentage.

	LP Continuous	GP+Round	MIGP
Oats	3.47	3	3
Honey	6.62	7	1
Banana	0.00	0	3
Dark chocolate chips	1.13	1	1
kcal	965.3 (-3.5%)	895.2 (-10.5%)	942.0 (-5.8%)
P (g)	25.0 (+0.0%)	21.7 (-13.2%)	25.4 (+1.6%)

Table 23: Example D: side-by-side method allocations. LP Continuous shows fractional servings (lower bound

z_{LP}

); GP+Round rounds each food to the nearest integer; MIGP finds the globally optimal integer solution.

Method	Feasible	Obj.	Max dev.	Cal	Prot	Carbs	Fat
MIGP	Yes	0.1437	-6.3%	-5.8	+1.5	-0.7	-6.3
GP+Round	Yes	0.4143	13.2%	10.5	13.2	5.0	12.8
Hard-IP	No	—	—	—	—	—	—

Table 24: Example D: method comparison with per-macro deviations (%).

Macro	% split	kcal/g	Calculation	Target
Calories	—	—	$600$	600.0 kcal
Protein	45%	4	$600\times 0.45\;/\;4=67.5$	67.5 g
Carbs	30%	4	$600\times 0.30\;/\;4=45.0$	45.0 g
Fat	25%	9	$600\times 0.25\;/\;9=16.7$	16.7 g

Table 25: Example E: target derivation from 600 kcal with 45/30/25 (P/C/F%).

			Per 100 g				Per serving
Food	Srv (g)	Range	kcal	P	C	F	kcal	P	C	F
Whey protein powder	30	[1, 2]	400	80.0	10.0	3.3	120.0	24.0	3.0	1.0
Tuna (canned)	80	[0, 3]	116	25.5	0.0	0.8	92.8	20.4	0.0	0.6
Greek yogurt	150	[0, 2]	59	10.2	3.6	0.7	88.5	15.3	5.4	1.1
White rice	50	[0, 6]	130	2.7	28.2	0.3	65.0	1.4	14.1	0.1
Banana	120	[0, 2]	89	1.1	22.8	0.3	106.8	1.3	27.4	0.4

Table 26: Example E: food inputs. Target: 600 kcal, 45/30/25 (P/C/F%). Derived gram targets: 67.5g protein, 45.0g carbs, 16.7g fat.

Food (serving)	Qty	kcal	P (g)	C (g)	F (g)
Whey protein powder (30 g)	1	120.0	24.0	3.0	1.0
Tuna (canned) (80 g)	2	185.6	40.8	0.0	1.3
White rice (50 g)	1	65.0	1.4	14.1	0.1
Banana (120 g)	1	106.8	1.3	27.4	0.4
Olive oil (5 g)	3	132.6	0.0	0.0	15.0
Total		610.0	67.5	44.5	17.8

Table 27: Example E: MIGP solution breakdown. Each row shows one food’s contribution; the deviation row shows (actual

-

target) / target as a percentage.

	LP Continuous	GP+Round	MIGP
Whey protein powder	2.00	2	1
Tuna (canned)	0.83	1	2
White rice	0.97	1	1
Banana	0.92	1	1
Olive oil	2.74	3	3
kcal	600.0 (+0.0%)	637.2 (+6.2%)	610.0 (+1.7%)

Table 28: Example E: side-by-side method allocations. LP Continuous shows fractional servings (lower bound

z_{LP}

); GP+Round rounds each food to the nearest integer; MIGP finds the globally optimal integer solution.

Method	Feasible	Obj.	Max dev.	Cal	Prot	Carbs	Fat
MIGP	Yes	0.0959	+6.7%	+1.7	-0.0	-1.2	+6.7
GP+Round	Yes	0.2574	8.8%	6.2	5.3	5.5	8.8
Hard-IP	No	—	—	—	—	—	—

Table 29: Example E: method comparison with per-macro deviations (%).

Figure 6: Sensitivity to penalty weight scheme (medium-loose config, 30 instances). Objective values across schemes are not directly comparable : each scheme uses different penalty magnitudes

w_{m}

, so a lower objective under equal weights does not imply a better solution than under inverse-target weights. The meaningful comparison is in per-macro deviation percentages: inverse-target and double-protein achieve balanced deviations (all median per-macro deviations under 1%); equal weights allow fat to deviate up to 2.4% (median).

Serving size	Median obj.	Max dev. %	Within 5%	Median time
25 g	0.001	0.1%	100%	2651 ms
50 g	0.008	0.3%	100%	131 ms
100 g	0.039	1.9%	100%	50 ms
200 g	0.108	6.2%	75%	30 ms

Table 30: Effect of serving granularity on solution quality and solve time. Finer granularity improves accuracy but increases computational cost. The 100 g default offers a practical tradeoff.

Mixed-Integer Linear Programming

Mixed-integer linear programming (MILP) optimizes linear objectives and constraints when some decisions must be integer-valued.

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