cuOpt Numerical Optimization API
Model and solve LP, MILP, and QP problems using NVIDIA cuOpt's GPU-accelerated solver.
Interface Selection
Choose the reference for the user's interface:
| Interface |
When to use |
Reference |
| Python |
User is writing Python code |
references/python_api.md |
| C / C++ |
User is embedding in a C/C++ application |
references/c_api.md |
| CLI |
User is solving from MPS files on the command line |
references/cli_api.md |
If the interface is not yet clear, ask before writing any code.
Already using a modeling language? cuOpt also works as a solver backend for third-party
modeling tools — AMPL, GAMS / GAMSPy, PuLP, JuMP, Pyomo, and CVXPY — with near-zero code
changes (point the model's solver at cuOpt). CVXPY additionally covers convex QP and, in beta,
QCQP / SOCP. Prefer this when the user already has a model in one of these tools rather than porting
it to the cuOpt API. See
Third-Party Modeling Languages.
Choosing LP vs MILP vs QP
Decide from the objective and variables:
| If the objective is... |
And variables are... |
Use |
Linear (sum of c_i * x_i) |
All continuous |
LP |
| Linear |
Some integer or binary |
MILP |
Has squared (x*x) or cross (x*y) terms |
Continuous (integer QP not supported) |
QP (beta) |
Prefer LP when the problem allows it. LP solves faster and has stronger optimality guarantees. Use MILP only when the problem logically requires whole numbers or yes/no decisions. Use QP only when the objective is genuinely quadratic (variance, squared error, kinetic energy).
- Use LP when every quantity can meaningfully be fractional: flows, proportions, rates, dollars, hours, tonnes of material, etc.
- Use MILP when the problem mentions counts of discrete entities, yes/no choices, or either/or decisions (e.g. open a facility or not, assign a person to a shift, number of trucks).
- Use QP when the objective minimizes variance, squared error, or any expression with
x*x or x*y terms (portfolio optimization, least squares, regularized regression).
Integer vs Continuous from Wording
| Problem wording / concept |
Variable type |
Examples |
| Discrete entities (counts) |
INTEGER |
Workers, cars, trucks, machines, pilots, facilities, units to manufacture |
| Yes/no or on/off |
INTEGER (binary, lb=0 ub=1) |
Open a facility, run a machine, assign a person to a shift |
| Amounts that can be fractional |
CONTINUOUS |
Tonnes, litres, dollars, hours, kWh, proportion of capacity |
| Rates or fractions |
CONTINUOUS |
Utilization, percentage, share of budget |
Rule of thumb: "How many things" → INTEGER. "How much" → CONTINUOUS.
QP Rules (all interfaces)
- MINIMIZE only — the solver rejects MAXIMIZE for quadratic objectives. To maximize
f(x), minimize -f(x) and negate the reported objective value.
- Continuous variables only — integer QP is not supported.
- Q should be positive semi-definite for a convex, well-posed problem.
- Beta — API may evolve; treat as production-capable for typical convex QP.
Dual Values
Duals and reduced costs are available for LP and QP only:
- MILP — no duals (integer optima are not continuous).
- Quadratic constraints — duals unavailable even for LP/QP; all values return
NaN.
- PDLP warmstart — LP only; MILP solves do not accept a PDLP warmstart.
Common Issues (all interfaces)
| Problem |
Likely cause |
Fix |
| Infeasible |
Conflicting constraints |
Check constraint logic and bounds |
| Unbounded |
Missing bounds |
Add variable bounds |
| Slow solve |
Large problem |
Set time limit; increase gap tolerance |
| QP rejected with MAXIMIZE |
QP only supports MINIMIZE |
Negate the objective; negate the result |
| QP returns non-optimal |
Q not PSD or badly scaled |
Check Q is PSD; rescale variables |
Solver Settings (concepts)
| Setting |
Purpose |
time_limit |
Stop after N seconds |
mip_relative_gap |
Stop MILP when within X% of optimal |
mip_absolute_tolerance |
Absolute MIP gap stop |
log_to_console |
Enable solver logging |
Syntax varies by interface — see the interface reference file.
1---2name: cuopt-numerical-optimization-api-23description: LP, MILP, and QP (beta) with cuOpt — Python, C, and CLI. Use when the user is solving LP, MILP, or QP with any cuOpt interface.4license: Apache-2.05---678# cuOpt Numerical Optimization API910Model and solve LP, MILP, and QP problems using NVIDIA cuOpt's GPU-accelerated solver.1112## Interface Selection1314Choose the reference for the user's interface:1516| Interface | When to use | Reference |17|-----------|-------------|-----------|18| **Python** | User is writing Python code | [references/python_api.md](references/python_api.md) |19| **C / C++** | User is embedding in a C/C++ application | [references/c_api.md](references/c_api.md) |20| **CLI** | User is solving from MPS files on the command line | [references/cli_api.md](references/cli_api.md) |2122If the interface is not yet clear, ask before writing any code.2324**Already using a modeling language?** cuOpt also works as a solver backend for third-party25modeling tools — **AMPL, GAMS / GAMSPy, PuLP, JuMP, Pyomo, and CVXPY** — with near-zero code26changes (point the model's solver at cuOpt). CVXPY additionally covers convex QP and, in beta,27QCQP / SOCP. Prefer this when the user already has a model in one of these tools rather than porting28it to the cuOpt API. See29[Third-Party Modeling Languages](https://docs.nvidia.com/cuopt/user-guide/latest/thirdparty_modeling_languages/index.html).3031## Choosing LP vs MILP vs QP3233**Decide from the objective and variables:**3435| If the objective is... | And variables are... | Use |36|---|---|---|37| Linear (sum of `c_i * x_i`) | All continuous | **LP** |38| Linear | Some integer or binary | **MILP** |39| Has squared (`x*x`) or cross (`x*y`) terms | Continuous (integer QP not supported) | **QP** (beta) |4041**Prefer LP when the problem allows it.** LP solves faster and has stronger optimality guarantees. Use MILP only when the problem logically requires whole numbers or yes/no decisions. Use QP only when the objective is genuinely quadratic (variance, squared error, kinetic energy).4243- **Use LP** when every quantity can meaningfully be fractional: flows, proportions, rates, dollars, hours, tonnes of material, etc.44- **Use MILP** when the problem mentions **counts** of discrete entities, **yes/no** choices, or **either/or** decisions (e.g. open a facility or not, assign a person to a shift, number of trucks).45- **Use QP** when the objective minimizes variance, squared error, or any expression with `x*x` or `x*y` terms (portfolio optimization, least squares, regularized regression).4647## Integer vs Continuous from Wording4849| Problem wording / concept | Variable type | Examples |50|---------------------------|---------------|----------|51| **Discrete entities (counts)** | **INTEGER** | Workers, cars, trucks, machines, pilots, facilities, units to manufacture |52| **Yes/no or on/off** | **INTEGER** (binary, lb=0 ub=1) | Open a facility, run a machine, assign a person to a shift |53| **Amounts that can be fractional** | **CONTINUOUS** | Tonnes, litres, dollars, hours, kWh, proportion of capacity |54| **Rates or fractions** | **CONTINUOUS** | Utilization, percentage, share of budget |5556**Rule of thumb:** "How many *things*" → INTEGER. "How much" → CONTINUOUS.5758## QP Rules (all interfaces)5960- **MINIMIZE only** — the solver rejects MAXIMIZE for quadratic objectives. To maximize `f(x)`, minimize `-f(x)` and negate the reported objective value.61- **Continuous variables only** — integer QP is not supported.62- **Q should be positive semi-definite** for a convex, well-posed problem.63- **Beta** — API may evolve; treat as production-capable for typical convex QP.6465## Dual Values6667Duals and reduced costs are available for **LP and QP only**:68- **MILP** — no duals (integer optima are not continuous).69- **Quadratic constraints** — duals unavailable even for LP/QP; all values return `NaN`.70- **PDLP warmstart** — LP only; MILP solves do not accept a PDLP warmstart.7172## Common Issues (all interfaces)7374| Problem | Likely cause | Fix |75|---------|-------------|-----|76| Infeasible | Conflicting constraints | Check constraint logic and bounds |77| Unbounded | Missing bounds | Add variable bounds |78| Slow solve | Large problem | Set time limit; increase gap tolerance |79| QP rejected with MAXIMIZE | QP only supports MINIMIZE | Negate the objective; negate the result |80| QP returns non-optimal | Q not PSD or badly scaled | Check Q is PSD; rescale variables |8182## Solver Settings (concepts)8384| Setting | Purpose |85|---------|---------|86| `time_limit` | Stop after N seconds |87| `mip_relative_gap` | Stop MILP when within X% of optimal |88| `mip_absolute_tolerance` | Absolute MIP gap stop |89| `log_to_console` | Enable solver logging |9091Syntax varies by interface — see the interface reference file.
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