Native PUBO
Represent sparse polynomial terms directly, including cubic and higher-order interactions.
Private API preview
Momentum Solve
Optimize constrained polynomial objectives in their native form. No quadratization. No auxiliary-variable inflation.
- 9/9
- Exact optima on small cubic trials
- 1,000
- Variables in published cubic study
- 60 s
- Matched CPU benchmark budget
Objective-
Selected0 / 12
Iteration0
Published evidence
Built for the structure other solvers remove.
HAMD searches native cubic interactions directly. The accompanying study compares decoded native objectives under equal time budgets.
VariablesImprovement vs SA / tabuBudget
20087.9%60 sec CPU
30071.2%60 sec CPU
50059.5%60 sec CPU
1,00046.9%60 sec CPU
Single-seed relative improvement in decoded native cubic objective; lower is better. See arXiv:2603.15947 for methodology and limitations.
Three fields. One native engine.
Complex structure goes in.
Decisions come out.
Express domain interactions as sparse polynomial terms, preserve hard selection counts, and solve through one asynchronous interface.
01
Cubic portfolio construction
Select exactly k assets while modeling returns, pairwise covariance, and three-way sector co-movement directly. Native cubic search avoids the auxiliary variables introduced by quadratization.
ReturnPair riskTriple interaction
Published outcome195.65median native objectiveSA / tabu1,208.07same 60-second budget
Measured at n=200 across three seeds
{
"problem_type": "pubo",
"objective": {
"num_vars": 200,
"terms": [
{ "vars": [12], "coef": -0.84 },
{ "vars": [12, 47], "coef": 1.31 },
{ "vars": [12, 47, 88], "coef": 2.76 }
]
},
"constraints": { "cardinality": 20 },
"time_limit_s": 60
}Benchmark result
{ "objective": 195.65, "feasible": true, "selected_count": 20 }02
Constrained route planning
Encode city-position assignments, travel costs, and one-hot feasibility penalties in a QUBO. The same search pipeline produces a valid tour before a short 2-opt polish pass.
HAMD + 2-opt4,517.7tour length at 30 cities2-opt baseline4,738.54.7% longer
100% feasibility in the published suite
{
"problem_type": "qubo",
"objective": {
"num_vars": 900,
"terms": [
{ "vars": [0, 31], "coef": 18.4 },
{ "vars": [0, 1], "coef": 80.0 },
{ "vars": [31], "coef": -80.0 }
]
},
"constraints": { "cardinality": 30 },
"time_limit_s": 60
}Benchmark result
{ "tour_length": 4517.7, "feasible": true, "cities": 30 }03
Higher-order constraint satisfaction
Map each three-literal clause to a native cubic penalty. Satisfied clauses contribute zero penalty; violated clauses raise the objective without introducing auxiliary bits.
x4OR¬x19ORx63SAT
Best gap0.00%at 75 variablesClauses satisfied100%published test instance
Matched the best baseline outcome
{
"problem_type": "pubo",
"objective": {
"num_vars": 75,
"terms": [
{ "vars": [4], "coef": -1.0 },
{ "vars": [4, 19], "coef": 1.0 },
{ "vars": [4, 19, 63], "coef": -1.0 }
]
},
"time_limit_s": 30,
"seed": 42
}Benchmark result
{ "gap": 0.0, "satisfied": true, "variables": 75 }04
No GPU required
The exact same HAMD dynamics code, unmodified, run end-to-end on a $0.083/hour 2-vCPU cloud instance — no CUDA, no accelerator. Solution quality tracks the GPU run within a few points; only wall-clock time changes.
CPU avg ratio95.8%of best-known, 6 Gset graphsGPU avg ratio95.9%same graphs, A10G
AWS t3.large · 2 vCPU · 7.6 GB RAM · zero GPU
| Graph | N | BKV | CPU best | CPU ratio | GPU best | GPU ratio |
|---|---|---|---|---|---|---|
| G1 | 800 | 11,624 | 11,449 | 98.5% | 11,467 | 98.6% |
| G11 | 800 | 564 | 530 | 94.0% | 520 | 92.2% |
| G14 | 800 | 3,064 | 2,982 | 97.3% | 3,002 | 98.0% |
| G51 | 1,000 | 3,848 | 3,749 | 97.4% | 3,775 | 98.1% |
| G22 | 2,000 | 13,359 | 12,982 | 97.2% | 13,034 | 97.6% |
| G32 | 2,000 | 1,410 | 1,278 | 90.6% | 1,282 | 90.9% |
Cost basis
{ "instance": "t3.large", "hourly_usd": 0.0832, "gpu_required": false }JSON arrays show representative sparse terms, not complete benchmark datasets. Portfolio results: arXiv:2603.15947. Routing and 3-SAT results: published Community Edition benchmark suite. CPU vs. GPU Max-Cut comparison: HAMD Community Edition bench_suite/benchmark_maxcut.py run unmodified on AWS t3.large (CPU) and AWS g5.xlarge / A10G (GPU).
One interface, complex objectives
Submit the math you mean.
QUBO compatible
Use the same job API for standard quadratic unconstrained binary objectives.
Exact cardinality
Preserve an exact number of selected variables throughout the search pipeline.
Asynchronous jobs
Submit, poll, and retrieve results without holding an open compute connection.
API preview
From polynomial to solution in two calls.
Send sparse coefficients and constraints. Momentum Solve returns a job identifier, then a selected-variable solution with its objective value.
POST /v1/jobs
GET /v1/jobs/{job_id}
Get a preview key
request.json
{
"problem_type": "pubo",
"objective": {
"num_vars": 48,
"terms": [
{ "vars": [0, 7, 21], "coef": -4.8 },
{ "vars": [3, 19], "coef": 1.25 }
]
},
"constraints": { "cardinality": 12 },
"time_limit_s": 30
}Open research, private production engine